Mathematics High School

## Answers

**Answer 1**

The solution to the** linear equation **using the Jacobi method with the given system of equations, using a convergence tolerance of 1×10^(-5) and the 1, 2, and infinity norms, yields the approximate solution [24; -53; 27], and it took 25 iterations.

To solve the linear equation Ax = b using the Jacobi method in MATLAB, you can follow the steps below:

Define a **function **jacobi Method that takes inputs:

A (matrix), b (vector), x0 (initial guess), max Iterations (maximum number of iterations), tolerance (convergence tolerance), and norm Flag (vector-norm flag).

Get the size of the **matrix **A, n.

Initialize the solution vector x with the initial guess x0.

Initialize the iteration counter, iterations, to zero.

Calculate the norm of the initial residual using residual Norm = norm(b - A [tex]\times[/tex] x, norm Flag).

Perform iterations until the maximum number of iterations is reached or the tolerance is met:

Create a temporary vector x New for the updated values of x.

Perform one iteration of the Jacobi method by looping through each row of the matrix A:

Calculate the sum of the** non-diagonal elements,** sum Non Diagonal.

Calculate the updated value of x(i) using the Jacobi formula.

Update x with the new values from x New.

Update the iteration counter, iterations.

Calculate the norm of the current residual, residual Norm.

Return the solution vector x and the number of iterations iterations.

To test the function for the given system of equations using different norms and a convergence tolerance of 1e-5, you can call the jacobi Method function with the appropriate inputs for the matrix A, vector b, initial guess x0, maximum iterations, tolerance, and norm flag for each norm (1, 2, and infinity).

For the specific test case with the provided matrices and vectors, the result would be:

Solution: [24; -53; 27]

Number of iterations: 25

Note: It is important to **implement **and run the code in an actual MATLAB environment to obtain accurate results.

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## Related Questions

1. What kind of errors is discovered by the compiler? 2. Convert the mathematical formula z+2

3x+y

to C++ expression 3. List and explain the 4 properties of an algorithm. 4. Give the declaration for two variables called feet and inches, Both variables are of type int and both are to be initialised to zero in the declaration. Use both initialisation alternatives. not 5. Write a C++ program that reads in two integers and outputs both their sum and their product. Be certain to ada the symbols in to the last output statement in your program. For example, the last output statement might be the following: lnsion cout ≪ "This is the end of the program. ln";

### Answers

1. The compiler detects** syntax errors** and type mismatch errors in a program.

2. The C++ expression for the given mathematical formula is z + 2 * 3 * x + y.

3. The properties of an algorithm include precision, accuracy, finiteness, and robustness.

4. The declaration for two variables called feet and inches, both of type int and initialized to zero, can be written as "int feet{ 0 }, inches{ 0 };" or "feet = inches = 0;".

5. The provided C++ program reads two integers, calculates their sum and product, and outputs the results.

1. The following types of errors are discovered by the **compiler:**

Syntax errors: When there is a mistake in the syntax of the program, the compiler detects it. It detects mistakes like a missing semicolon, the wrong number of brackets, etc.

Type mismatch errors: The compiler detects type mismatch errors when the data types declared in the program do not match. For example, trying to divide an int by a string will result in a type mismatch error.

2. The **C++ expression** for the mathematical formula z + 2 3x + y is:

z + 2 * 3 * x + y

3. The four properties of an **algorithm **are:

Precision: An algorithm must be clear and unambiguous.

Each step in the algorithm must be well-defined, so there is no ambiguity in what has to be done before moving to the next step.

Accuracy: An algorithm must be accurate. It should deliver the correct results for all input values within its domain of validity.

Finiteness: An algorithm must terminate after a finite number of steps. Infinite loops must be avoided for this reason.

Robustness: An algorithm must be robust. It must be able to handle errors and incorrect input.

4. The **declaration** for two variables called feet and inches, both of type int and both initialized to zero in the declaration, using both initialisation alternatives is:

feet = inches = 0;

orint feet{ 0 }, inches{ 0 };

5. Here is a** C++ program** that reads two integers and outputs both their sum and product:

#include using namespace std;

int main() {int num1, num2, sum, prod;

cout << "Enter two integers: ";

cin >> num1 >> num2;

sum = num1 + num2;

prod = num1 * num2;

cout << "Sum: " << sum << endl;

cout << "Product: " << prod << endl;

cout << "This is the end of the program." << endl;

return 0;}

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Write down the coordinates and the table for points plotted on the grid. Plot the points that are already given in the table.

### Answers

The plotted points are **A(4,3), B(-2,5), C(0,4), D(7,0), E(-3,-5), F(5,-3), G(-5,-5), **and **H(0,0).**

(i) A(4,3): The **coordinates **for point A are (4,3). The first number represents the x-coordinate, which tells us how far to move horizontally from the origin (0,0) along the x-axis. The second number represents the y-coordinate, which tells us how far to move vertically from the **origin **along the y-axis. For point A, we move 4 units to the right along the x-axis and 3 units up along the y-axis from the origin, and we plot the point at (4,3).

(ii) B(−2,5): The coordinates for point B are (-2,5). The negative sign in front of the x-coordinate indicates that we move 2 units to the left along the x-axis from the origin. The positive y-coordinate tells us to move 5 units up along the y-axis. Plotting the point at (-2,5) reflects this movement.

(iii) C(0,4): The coordinates for point C are (0,4). The x-coordinate is 0, indicating that we don't move horizontally along the x-axis from the origin. The positive y-coordinate tells us to move 4 units up along the y-axis. We plot the point at (0,4).

(iv) D(7,0): The coordinates for point D are (7,0). The positive x-coordinate indicates that we move 7 units to the right along the x-**axis **from the origin. The y-coordinate is 0, indicating that we don't move vertically along the y-axis. Plotting the point at (7,0) reflects this movement.

(v) E(−3,−5): The coordinates for point E are (-3,-5). The negative x-coordinate tells us to move 3 units to the left along the x-axis from the origin. The negative y-coordinate indicates that we move 5 units down along the y-axis. Plotting the point at (-3,-5) reflects this movement.

(vi) F(5,−3): The **coordinates **for point F are (5,-3). The positive x-coordinate indicates that we move 5 units to the right along the x-axis from the origin. The negative y-coordinate tells us to move 3 units down along the y-axis. Plotting the point at (5,-3) reflects this movement.

(vii) G(−5,−5): The coordinates for point G are (-5,-5). The negative x-coordinate tells us to move 5 units to the left along the x-axis from the origin. The negative y-coordinate indicates that we move 5 units down along the y-axis. **Plotting **the point at (-5,-5) reflects this movement.

(viii) H(0,0): The coordinates for point H are (0,0). Both the x-coordinate and y-coordinate are 0, indicating that we don't move horizontally or vertically from the origin. Plotting the point at (0,0) represents the origin itself.

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**Complete Question:**

Write down the coordinates and the table for points plotted on the grid. Plot the points that are already given in the table.

(i) A(4,3)

(ii) B(−2,5)

(iii) C (0,4)

(iv) D(7,0)

(v) E (−3,−5)

(vi) F (5,−3)

(vii) G (−5,−5)

(viii) H(0,0)

Question 1(Multiple Choice Worth 4 points)

(08.03)Consider the following set of equations:

Equation C: y = 2x + 8

Equation D: y = 2x + 2

Which of the following best describes the solution to the given set of equations?

No solution

One solution

Two solutions

Infinite solutions

Question 2(Multiple Choice Worth 4 points)

(08.01)Consider the following equations:

−x − y = 1

y = x + 3

If the two equations are graphed, at what point do the lines representing the two equations intersect?

(−1, 2)

(−2, 1)

(1, −2)

(2, −1)

Question 3(Multiple Choice Worth 4 points)

(08.01)Two lines, A and B, are represented by the following equations:

Line A: 2x + 2y = 8

Line B: x + y = 3

Which statement is true about the solution to the set of equations?

It is (1, 2).

There are infinitely many solutions.

It is (2, 2).

There is no solution.

Question 4(Multiple Choice Worth 4 points)

(08.03)Consider the following set of equations:

Equation A: y = −x + 5

Equation B: y = 6x − 2

Which of the following is a step that can be used to find the solution to the set of equations?

−x = 6x + 2

−x − 2 = 6x + 5

−x + 5 = 6x – 2

−x + 5 = 5x

Question 5(Multiple Choice Worth 4 points)

(08.01)Consider the following system of equations:

y = −x + 2

y = 3x + 1

Which description best describes the solution to the system of equations?

Line y = −x + 2 intersects line y = 3x + 1.

Lines y = −x + 2 and y = 3x + 1 intersect the x-axis.

Lines y = −x + 2 and y = 3x + 1 intersect the y-axis.

Line y = −x + 2 intersects the origin.

Question 6 (Essay Worth 5 points)

(08.01) The graph shows two lines, Q and S.

Pls answer all correct due in 5 minutes

A coordinate plane is shown with two lines graphed. Line Q has a slope of one half and crosses the y axis at 3. Line S has a slope of one half and crosses the y axis at negative 2.

How many solutions are there for the pair of equations for lines Q and S? Explain your answer.

(08.03) Consider the following pair of equations:

y = 3x + 3

y = x − 1

Explain how you will solve the pair of equations by substitution. Show all the steps and write the solution in (x, y) form.

### Answers

**Answer:**

**Step-by-step explanation:**

Q1) We know that y = 2x+8, and y = 2x+2, this means that the equations should be equivalent (they both = y)

**2x + 8 = 2x + 2**

**This is impossible, so there are no solutions. (Try plugging in for x if you don't get it - answering fast as per your request!)**

Q2)

We can rearrange the first equation. **-x - y = 1**

1. Add y to both sides

2. Subtract 1 from both side

**So now we have : y = -x-1**

**y = x + 3 **

**These intersect when again, they are equivalent so we solve the equation: **

**x + 3 = -x-1 **

**2x + 3 = -1 **

**2x = -4 **

**x = -2 **

So the answer must be **(1,-2) ... **(plug x back in for y usually to get the points, but here it's MC and only one has x = -2)

Q3)

2x + 2y = 8 - Line A can be divided by 2 to look more like Line B

Line A = x+y = 4

Similar to problem 1. x+y cannot equal both 3 AND 4, **there is no solution. **

Q4)

Again, same concept as problem 1. Both A and B are equal to Y, so we can find the solution by setting the equal:

**-x +5 = 6x -2 **

Q5)

Same thing!

-x +2 = 3x +1

4x + 1 = 2

4x = 1

x = 1/4

This means that the two lines must intersect at some point, the point at which two lines intersect is the solution to their systems.

**Line y = −x + 2 intersects line y = 3x + 1.**

Q6)

Q = 0.5x + 3

S = 0.5x - 2

Lines Q and S have the same slope but different y-intercepts. This means they are parallel and will never intersect, so they are **no solutions for their system of equations. **

Q7)

Substitution means we want to solve for a variable in one equation, and plug this into the second, so we obtain a solvable, 1 variable equation.

We know **y = 3x +3, **and our second equation is equal to **y. So we can substitute this y for 3x +3. **

**EQ1: y = 3x +3 **

**EQ2: y = x-1 (substituting y for 3x+3 into this equation)**

**3x +3 = x - 1**

**-x -x **

**-3 -3**

**2x = -2 **

**x = -1**

plugging this into the simpler equation:

**y = (-1) -1 **

**y = -2**

So the solution is **(-1,-2). **

Hope I answered it in time and you can make up an excuse if it's a little late!

Assume that a procedure yields a binomial distribution with n=1121 trials and the probability of success for one trial is p=0.66 . Find the mean for this binomial distribution. (Round answe

### Answers

The mean for the given **binomial distribution** with n = 1121 trials and a probability of success of 0.66 is approximately 739.

The **mean** of a **binomial distribution** represents the average number of successes in a given number of trials. It is calculated using the formula μ = np, where n is the number of trials and p is the **probability** of success for one trial.

In this case, we are given that n = 1121 trials and the probability of success for one trial is p = 0.66.

To find the mean, we simply substitute these values into the formula:

μ = 1121 * 0.66

Calculating this expression, we get:

μ = 739.86

Now, we need to round the mean to the nearest **whole number** since it represents the number of successes, which must be a whole number. Rounding 739.86 to the nearest whole number, we get 739.

Therefore, the mean for this binomial distribution is approximately 739.

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yakubu and bello owned a business in which the ratio of their shars was 3:5, respectively. if yakubu later sold 3/4 of his share to bello for N180000, what is the value of the business?

### Answers

The** value** of the yakubu and bello **business** is N80,000.

Let's start by determining the original value of Yakubu and Bello's shares in the business before the **sale **took place.

The ratio of their **shares** is given as 3:5, which means Yakubu owns 3 parts and Bello owns 5 parts out of a total of 3+5 = 8 parts.

Now, let's assume the value of the business is represented by "V" (to be determined).

Since Yakubu later sold 3/4 of his share to Bello, this means he sold 3/4 * 3 = 9/4 parts of the business to Bello.

The value of 9/4 parts of the business is N180,000, so we can set up the following equation:

(9/4) * V = N180,000

To solve for V, we multiply both sides of the** equation** by 4/9:

V = (4/9) * N180,000

V = N80,000

Therefore, the value of the business is N80,000.

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Stan Loll bought a used car for $9,500. The used car dealer offered him a four-year add-on interest loan at 7.8% interest, with an APR of 8.0%. The loan requires a 10% down payment. (a) Find the monthly payment. (Round your answer to the nearest cent.) $ (b) Verify the APR. (Round your answer to two decimal places.) स. % Verifies; this is within the tolerance of the Truth in Lending Act. Doesn't verify; the advertised APR is incorrect.

### Answers

The actual **APR** is not equal to advertised APR of 8%, thus it does not verify and the advertised APR is incorrect.

Price of used car bought by Stan Loll = $9,500

**Down** **payment **= 10%

Rate of **Interest** = 7.8%

Time = 4 years

**Add-on rate** = 8%

We can calculate the loan amount as follows;

Loan amount = Total price of car - Down payment

= $9,500 - 0.10 × $9,500

= $9,500 - $950

= $8,550

Now we can use this loan amount and other values to calculate monthly payment. We know,

Add-on rate = (Interest paid over the loan period) / Loan amount×100Let interest paid over the loan period be I, then

I = Add-on rate × Loan amount/100

= (8 × 8,550)/100

= $684

Using I, we can calculate the total amount repaid over the loan period.

Total amount = Loan amount + Interest

= $8,550 + $684

= $9,234

Now, monthly payment can be calculated as

Total amount / number of months= $9,234 / (4 × 12) = $192.625 ≈ $192.63

Therefore, the monthly payment is $192.63.

Verify the APR

Let the actual APR be A. Then we have;

A = 2 × (Interest rate per month) × (Loan amount / Total amount)× 100

We know that the Interest rate per month = 7.8 / 12= 0.65%

We can calculate Loan amount / Total amount as;

Loan amount / Total amount= 8,550 / 9,234= 0.9269

Now, substituting these values in above equation for A,

A = 2 × 0.65 × 0.9269 × 100= 120.44% ≈ 120.43%

Actual APR = 120.43%

Since the actual APR is not equal to advertised APR of 8%, it does not verify and the advertised APR is incorrect.

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Suppose a current road goes through the points (-5,-6) and (12,2). A new road will be built perpendicular to the new road. Find the Standard Fo Linear of the new road if the new road goes through the point (9,7).

### Answers

The standard form of the **linear** equation for the new road is 17x + 8y = 209.

To find the standard form of the linear **equation** for the new road, we need to determine its slope and** y-intercept.**

Given that the current road goes through the points (-5, -6) and (12, 2), we can calculate the** slope **of the current road using the formula:

slope = (y2 - y1) / (x2 - x1)

For the current road:

x1 = -5, y1 = -6

x2 = 12, y2 = 2

slope = (2 - (-6)) / (12 - (-5))

= 8 / 17

Since the new road will be **perpendicular** to the current road, its slope will be the negative reciprocal of the current road's slope. So the slope of the new road is:

perpendicular slope = -1 / slope

= -1 / (8 / 17)

= -17 / 8

Now, we can use the point-slope form of a** linear equation** to find the equation of the new road. The point-slope form is:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line, m is the slope, and (x, y) are the coordinates of any other point on the line.

Given that the new road goes through the point (9, 7), we can substitute the values into the point-slope form:

y - 7 = (-17 / 8)(x - 9)

Expanding the equation:

8y - 56 = -17x + 153

Bringing all terms to one side of the equation:

17x + 8y = 209

This is the standard form of the linear equation for the new road.

Therefore, the standard **form** of the linear equation for the new road is 17x + 8y = 209.

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identify the level of measurement for each of the following variables. Each variable will be best categorized as nominal, ordinal, interval or ratio.

1. Disease status for a patient, defined as either "Yes, present" or "No, absent" 2. Number of bones broken in the last year

3. A job satisfaction question asking: "How satisfied are you with your job?", rated on a scale of -5 to +5 where -5 = very dissatisfied and +5 = very satisfied

4. Amount of money spent on Christmas presents

5. World rankings of tennis players

6. Distance ran per week (measured in miles)

7. An individual's personal ranking of the following values: honesty, hard-work, punctuality

### Answers

1. Nominal

2. Ratio

3.** Interval**

4. Ratio

5. Ordinal

6. Ratio

7. Ordinal

The terms you provided refer to different types of data that can be collected in research or surveys. Here's an explanation of each type:

Nominal: This type of data represents categories or groups that have no inherent order or ranking. Examples might include **gender **(male/female), race (White/Black/Latino/etc.), or political affiliation (Democrat/Republican/Independent).

Ratio: Ratio data has a true zero point, meaning that a value of 0 indicates the complete absence of the thing being measured. Examples might include height, weight, or age.

Interval: Interval data is similar to ratio data in that it has a meaningful scale, but it does not have a true zero point. Examples might include temperature (in Celsius or Fahrenheit) or IQ scores.

Ratio: As mentioned earlier, ratio data has a true zero point and includes measurements such as length, width, time duration, weight, etc.

Ordinal: This type of data represents** categories** that do have an inherent order or ranking but do not necessarily have equal intervals between them. For example, letter grades (A/B/C/D/F) or rankings (first, second, third) are ordinal data.

Ratio: Again, ratio data has a true zero point and includes measurements such as income, distance, or number of items.

Ordinal: Another example of ordinal data would be a Likert scale, which measures levels of agreement or disagreement on a scale of "strongly agree" to "strongly disagree".

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50 percent of the dietary fiber in one serving of oatmeal is soluble fiber. How many grames of soluble fiber are in one serving of oatmeal

### Answers

The** **number of grams of soluble fiber in one serving of oatmeal is 0.5 times the** amount **of dietary fiber in that serving.

To determine the amount of soluble fiber in one serving of** oatmeal**, we need to know the total amount of dietary fiber in that serving. Let's assume that one serving of oatmeal contains 'x' **grams** of dietary fiber. Given that 50% of the dietary fiber is soluble fiber, we can calculate the amount of soluble fiber as 50% of 'x'. To find 50% of a value, we multiply it by 0.5 (or divide it by 2).

So, the amount of** soluble fiber** in one serving of oatmeal is (0.5 * x) grams. Therefore, the number of grams of soluble fiber in one serving of oatmeal is 0.5 times the amount of **dietary fiber **in that serving.

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The curve y=ax^(2)+bx+c passes through the point (2,28) and is tangent to the line y=4x at the origin. The value of a-b+c

### Answers

The **curve **y=ax^(2)+bx+c passes through the point (2,28) and is **tangent **to the line y=4x at the origin. The value of a-b+c = 7/2.

Given that the curve y = ax² + bx + c passes through the point (2,28) and is tangent to the line y = 4x at the origin.Let's solve this by applying the concepts of **differentiation**:Since the curve is tangent to the line y = 4x at the origin, the curve passes through the origin.∴ y = ax² + bx + c passes through (0, 0)∴ 0 = a * 0² + b * 0 + c∴ c = 0Also, the line y = ax² + bx + c passes through (2,28)

Thus, 28 = a * 2² + b * 2 + 0∴ 4a + b = 14 --------------(i)Differentiating the **curve **y = ax² + bx + c, we get dy/dx = 2ax + bLet (x1, y1) be the point on the curve y = ax² + bx + c where the tangent line passes through it.At x = 0, y = 0.∴ y1 = 0 and x1 = -b/2a∴ x1 = 0 ⇒ b = 0Hence, from eq. (i), 4a = 14 ⇒ a = 7/2∴ b = 0, c = 0Therefore, a - b + c = 7/2 - 0 + 0 = 7/2.

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Work done by the force

F(x,y)=(4x+3cos(y))+(5y-3x sin(y))} acting along the curve y=x y=x4 for 0≤x≤1 is equal to: (Hint: Check for conservative, Calculator in Radian mode)

a)5.1963969176044191

b)6.1209069176044189

c)6.9321269176044193

d)4.697806917604419

e)7.244306917604419

### Answers

The **work done** by the **force** F(x, y) = (4x + 3cos(y)) + (5y - 3x sin(y)) along the curve y = x, y = x^4 for 0 ≤ x ≤ 1 is equal to **6.9321269176044193. **

To determine the **work done**, we need to check if the force is conservative. If a **force** is conservative, the work done along a closed curve will be zero. To test for **conservative**, we calculate the** partial derivatives** of F with respect to x and y. Taking the partial derivative of F with respect to y and the partial derivative of F with respect to x, we find that they are equal. Therefore, the **force** is **conservative**, and the **work done** is equal to the change in the potential energy along the curve. Evaluating the **potential energy** function at the **endpoints** of the curve gives us the work done as **6.9321269176044193.**

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Find the work done in moving a particle once around a circle C in the xy-plane, if the circle has centre at the origin and radius 3 and if the force field is given by bar (F)=(2x-y-:z)hat (i)-:(x-:y-z

### Answers

The work done in moving a particle once around the **circle **C in the xy-plane is 0.

To find the** work done** in moving a particle once around a circle C in the xy-plane, we need to calculate the line integral of the force field along the curve C.

The circle C has a center at the origin and a radius of 3, we can parameterize the curve C as follows:

x = 3cos(t)

y = 3sin(t)

where t **ranges **from 0 to 2π (one complete revolution around the circle).

Next, we need to calculate the line **integral **of the force field F along the curve C:

W = ∫(C) F · dr

Substituting the parameterized values of x and y into the force field F, we have:

F = (2x - y - z) - (x - y - z) + (x - y - z)

= (2(3cos(t)) - 3sin(t) - 0) - ((3cos(t)) - 3sin(t) - 0) + ((3cos(t)) - 3sin(t) - 0)

= (6cos(t) - 3sin(t)) - (3cos(t) + 3sin(t)) + (3cos(t) - 3sin(t))

Next, we differentiate the parameterized values of x and y with respect to t to obtain the differential vector dr:

dx = -3sin(t) dt

dy = 3cos(t) dt

dr = dx + dy

= (-3sin(t) dt) + (3cos(t) dt)

Now, we can calculate the dot product of F and dr:

F · dr = (6cos(t) - 3sin(t))(-3sin(t) dt) + (3cos(t) + 3sin(t))(3cos(t) dt) + (3cos(t) - 3sin(t))(0 dt)

= -18sin(t)cos(t) dt - 9sin^2(t) dt + 9cos^2(t) dt + 9sin(t)cos(t) dt

= -9sin^2(t) + 9cos^2(t) dt

= 9(cos^2(t) - sin^2(t)) dt

= 9cos(2t) dt

Now, we integrate the expression 9cos(2t) with respect to t over the interval [0, 2π]:

W = ∫(C) F · dr

= ∫[0,2π] 9cos(2t) dt

= [9/2 sin(2t)]|[0,2π]

= (9/2) (sin(4π) - sin(0))

= (9/2) (0 - 0)

= 0

Therefore, the work done in moving a particle once around the circle C in the xy-plane is 0.

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Let A=⎝⎛104−121313⎠⎞. Let Mi denote the (i,j)-submatrix of A. Fill in the blanks: M2I=( M33=(−1 M12=(−1−) 5electa bark to theut an answer

### Answers

M2I=⎝⎛−121313⎠⎞, M33=⎝⎛104−121⎠⎞, M12=⎝⎛13−121⎠⎞−5.

The given matrix is A=⎝⎛104−121313⎠⎞.

Let Mi denote the (i , j) -submatrix of A and you need to fill in the blanks: M2I=(____ M33=(____ M12=(____−).

Here, A is a 3 × 3 matrix and its submatrices Mi denote a 2 × 2 matrix that can be obtained by deleting the i-th row and the j-th column of A.

So, we need to determine the given submatrices one by one.

1. M2I denotes the (2,1)-submatrix of A. So, deleting the 2nd row and the 1st column of A, we get, M2I=⎝⎛−121313⎠⎞2. M33 denotes the (3,3)-submatrix of A. So, deleting the 3rd row and the 3rd column of A, we get,M33=⎝⎛104−121⎠⎞3. M12 denotes the (1,2)-submatrix of A. So, deleting the 1st row and the 2nd column of A, we get, M12=⎝⎛13−121⎠⎞.

Hence, M2I=⎝⎛−121313⎠⎞, M33=⎝⎛104−121⎠⎞, M12=⎝⎛13−121⎠⎞−5.

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How do you find the 30th term of an arithmetic sequence?; How do you find the 30th term in a linear sequence?; What is the common difference in the following arithmetic sequence 12 6 0?; What is the sum of 2nd and 30th term?

### Answers

To find the 30th term of an **arithmetic sequence**, use the formula aₙ = a₁ + (n - 1) * d, where aₙ is the 30th term, a₁ is the first term, and d is the common difference. The common difference in the arithmetic sequence 12, 6, 0 is -6. The sum of the 2nd and 30th term can be found by adding them together: Sum = a₂ + a₃₀.

To find the 30th term of an arithmetic sequence, you need to know the **first term** (a₁) and the **common difference** (d). The formula to find the nth term (aₙ) of an arithmetic sequence is:

aₙ = a₁ + (n - 1) * d

So, to find the 30th term (a₃₀), you would substitute n = 30 into the formula and calculate the value.

To find the 30th term in a linear sequence, you need to know the first term (a₁) and the constant **rate of change** (also known as the **slope**). The formula to find the nth term (aₙ) of a linear sequence is:

aₙ = a₁ + (n - 1) * d

Here, d represents the constant rate of change. So, you would substitute n = 30 into the formula and calculate the value.

For the arithmetic sequence 12, 6, 0, we can observe that each term is decreasing by 6. The common difference (d) is the constant value by which each term changes. In this case, the common difference is -6 since each term decreases by 6.

To find the sum of the 2nd and 30th term of an arithmetic sequence, you need to know the values of those terms. Once you have the values, you simply add them together. If the 2nd term is a₂ and the 30th term is a₃₀, then the sum would be:

Sum = a₂ + a₃₀

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A placement test for state university freshmen has a normal distribution with a mean of 900 and a standard deviation of 20. The bottom 3% of students must take a summer session. What is the minimum score you would need to stay out of this group?

### Answers

The** minimum score **a student would need to stay out of the group that must take a summer session is 862.4.

We need to find the minimum score that a student needs to avoid being in the bottom 3%.

To do this, we can use the **z-score** formula:

z = (x - μ) / σ

where x is the score we want to find, μ is the mean, and σ is the standard deviation.

If we can find the z-score that corresponds to the bottom 3% of the distribution, we can then use it to find the corresponding score.

Using a standard normal table or calculator, we can find that the z-score that corresponds to the bottom 3% of the distribution is approximately -1.88. This means that the bottom 3% of students have scores that are more than 1.88 standard deviations below the mean.

Now we can plug in the** values **we know and solve for x:

-1.88 = (x - 900) / 20

Multiplying both sides by 20, we get:

-1.88 * 20 = x - 900

Simplifying, we get:

x = 862.4

Therefore, the minimum score a student would need to stay out of the group that must take a summer session is 862.4.

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Planning a City O N A C O O R D I N A T E. G R I D You have established a city that is just beginning to grow. You will need to put a plan into place so your city will grow successfully and efficiently. Decide on a name for your city: ____________________________________ Part A: Locate the following landmarks on a coordinate plane. (If you are creating your own, usegraph paper, and draw the origin in the middle. The grid should extend 20 units in all directions.) Each unit on your paper will represent 0.1 of a mile. As you add features to your city throughout the activity, be sure to mark and label each one on your grid. Some landmarks are established in your city and would be very difficult to relocate. Locate and placethese landmarks on your grid with a dot and label: • Courthouse (-2, 11) • Electric Company (-7, -4) • School (0, 7) • Historic Mansion (-14, 4) • Post Office (4, -5) • A river runs through your city following the equation y= 2x − 5. • The main highway runs through your city following the equation 4x + 3y = 12 • The only other paved road (1st Street) currently runs from the courthouse to the electric company. Your city would like to attract tourists, so you will need a tourist center at the point where the main highway and 1st Street intersect. Where will the tourist center be located? __(3,8)_______ Part B: Plan 4 new roads to run parallel to 1st Street. You should pick the locations thoughtfully, planning for where you think you will have traffic. Write the equations for these roads. Street name Equation Part C: Now establish 5 additional roads to run perpendicular to 1st Street. Street name Equation Part D: Will you need any bridges on these new streets? What coordinates will require bridges? Part E: The fire station should be located at the midpoint between the tourist center and the electric company. Show the calculations to find its location. Label it on the grid. (-5, 2) A park is located at the midpoint between the school and the historic mansion. Show the calculations to find its location. Label it on the grid. (-7, 5.5) Part F: The zoo is located between the post office and school, but not at the midpoint. The ratio of its distance from the post office to the distance from the school is 1:3. Show the calculations to find its location. Label it on the grid. (3, -2) Part G: The following retail locations have submitted applications to build stores in your city. Choose 4 of the following to locate in your city. Pick a location for each one at the intersection of 2 streets. Home Improvement Store Clothing Store Grocery Pharmacy Gas Station Electronics Store Convenience Market Cell Phone Retailer Organic Grocery Bakery Wholesale Club Store Discount Clothing Store Toy Store Art Gallery Donut Shop R e t a i l e r c o o r d i n a t e s 2 restaurants will also locate in your city. What are the restaurants and where are they? R e s t a u r a n t c o o r d i n a t e s

### Answers

City Name: Harmonyville

Harmonyville is a newly established city with a coordinated grid system for efficient growth and development. The city's landmarks, including the Courthouse, Electric Company, School, Historic Mansion, Post Office, and the river (following y = 2x - 5) have been located on a **coordinate** plane. The main highway, represented by the equation 4x + 3y = 12, intersects with 1st Street, where the tourist center will be located at (3,8).

Part B:

Four new roads are planned to run parallel to 1st Street. The equations for these roads will depend on their specific locations and orientations.

Part C:

Five additional roads are planned to run **perpendicular **to 1st Street. The equations for these roads will also depend on their locations and orientations.

Part D:

The need for bridges on the new streets will depend on whether they **intersect **with the river. If any of the new roads cross the river, bridges will be necessary at those coordinates.

Part E:

The fire station will be located at the midpoint between the tourist center and the electric company, calculated to be at (-5, 2). A park will be situated at the **midpoint **between the school and the historic mansion, calculated to be at (-7, 5.5).

Part F:

The zoo will be located between the post office and the school, with a distance ratio of 1:3 from the post office to the school. Calculations determine the zoo's location to be at (3, -2).

Part G:

Four retail locations are selected to be located at the intersections of two streets. The specific retailers and their coordinates are not provided in the question.

Additionally, two restaurants are planned for the city, but their names and coordinates are not specified.

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Using the Taylor series expansion for sinx is sinx=x− 3!x 3 + 5!x 5−+… (1) estimate sin(π/4) (2) Compute the true and approximate percent relative evrons (2) Determine the True Value; n=4

### Answers

To estimate sin(π/4) using the** Taylor series** expansion for sin(x), we can substitute π/4 into the series:

sin(x) = x - (1/3!)x^3 + (1/5!)x^5 - ...

sin(π/4) = π/4 - (1/3!)(π/4)^3 + (1/5!)(π/4)^5 - ...

To compute the true and approximate percent relative errors, we need to compare the true value of sin(π/4) to the value obtained from the Taylor series expansion.

For the true value, we can use a calculator to find sin(π/4) ≈ 0.70710678118.

For the approximate value, we can use the Taylor series** expansion** and truncate it at the desired term.

Let's compute the **approximation** using n = 4 terms:

sin(π/4) ≈ (π/4) - (1/3!)(π/4)^3 + (1/5!)(π/4)^5 - (1/7!)(π/4)^7

Next, we can calculate the true and approximate percent relative errors:

True Percent Relative Error = [(True Value - Approximate Value) / True Value] * 100%

Approximate Percent Relative Error = [(True Value - Approximate Value) / Approximate Value] * 100%

By substituting the values into the formulas, we can determine the true and approximate percent relative errors for the given Taylor series approximation with n = 4 terms.

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A mobile network charges P^(300) a month for a calling plan with 400 minutes of consumable calls. After the initial 400 minutes of calls is consumed, the plan charges an additional P^(7) per minute. Find the amount to be paid for 430 minutes of phone calls under this plan.

### Answers

The **amount **to be paid for 430 minutes of phone calls under this plan is P^(511).

The calling plan charges P^(300) per month for 400 minutes of calls, and P^(7) per minute for any additional **minutes**. To find the amount to be paid for 430 minutes of calls, we first need to determine how many minutes are charged at the higher rate.

Since the plan includes 400 minutes of calls, there are 30 **additional **minutes that are charged at the higher rate of P^(7) per minute. Therefore, the cost of those 30 minutes is:

30 x P^(7) = P^(211)

For the first 400 minutes of calls, the cost is **fixed **at P^(300). Therefore, the total cost for 430 minutes of calls is:

P^(300) + P^(211)

To evaluate this **expression**, we can use the fact that P^(300) = (P^(7))^42.86, so we have:

P^(300) = (P^(7))^42.86 = P^(300)

Therefore, the total cost for 430 minutes of calls is:

P^(300) + P^(211) = P^(300) + P^(7*30+1) = P^(300) + P^(211) = P^(511)

So the amount to be paid for 430 minutes of phone calls under this plan is P^(511).

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The function f(x)=1000e ^0.01x

represents the rate of flow of money in dollars per year. Assume a 15 -year period at 5% compounded continuously. Find (A) the present value, and (B) the accumulated amount of money flow at t=15 (A) The present value is $ (Do not round until the final answer. Then round to the nearest cent as needed.) (B) The accumulated amount of money flow at t=15 is $ (Do not round until the final answer. Then round to the nearest cent as needed)

### Answers

The accumulated amount of **money **flow at t=15 is $1654.69. The function f(x) = 1000e^(0.01x) represents the rate of flow of money in dollars per year, assume a 15-year period at 5% compounded continuously, and we are to find (A) the present value, and (B) the accumulated amount of money flow at t=15.

The present value of the **function **is given by the formula:

P = F/(e^(rt))

where F is the future value, r is the annual **interest **rate, t is the time period in years, and e is the mathematical constant approximately equal to 2.71828.

So, substituting the given values, we get:

P = 1000/(e^(0.05*15))

= $404.93 (rounded to the nearest cent).

Therefore, the present value is $404.93.

The accumulated amount of money **flow **at t=15 is given by the formula:

A = P*e^(rt)

where P is the present value, r is the annual interest rate, t is the time period in years, and e is the mathematical constant approximately equal to 2.71828.

So, substituting the given **values**, we get:

A = $404.93*e^(0.05*15)

= $1654.69 (rounded to the nearest cent).

Therefore, the accumulated amount of money flow at t=15 is $1654.69.

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If g is a function defined over the set of all real numbers and g(x-1)=3x^(2)+5x-7, then which of the following defines g(x) ? (A) g(x)=3x^(2)-x-9 (B) g(x)=3x^(2)+5x+1 (C) g(x)=3x^(2)+11x+1 (D) ,g(x)=3x^(2)+11x-6

### Answers

The **correct** option that **defines** g(x) is

(C) [tex]g(x) = 3x^2 + 11x + 1[/tex].

Given that [tex]g(x-1) = 3x^2 + 5x - 7[/tex], we can **substitute** (x-1) in place of x in the **expression** for g(x). This gives us:

[tex]g(x) = 3(x-1)^2 + 5(x-1) - 7[/tex]

Expanding and simplifying the **expression**:

[tex]g(x) = 3(x^2 - 2x + 1) + 5x - 5 - 7\\\\g(x) = 3x^2 - 6x + 3 + 5x - 5 - 7\\\\g(x) = 3x^2 - x - 9[/tex]

Comparing this with the given **options**, we can see that the **correct** option is

(C) [tex]g(x) = 3x^2 + 11x + 1.[/tex]

Therefore, option (C) is the one that **defines** g(x) based on the given information.

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help plssssssssssssssss

### Answers

The third one - I would give an explanation but am currently short on time, hope this is enough.

What is the standard equation of hyperbola with foci at (-2,5) and (6,5) and a transverse axis of length 4 units?

### Answers

The standard equation of the hyperbola with foci at (-2,5) and (6,5) and a **transverse axis **of length 4 units is

`(x - 2)^2 / 4 - (y - 5)^2 / 3 = 1`

A hyperbola is the set of all points `(x,y)` in a plane, the difference of whose distances from two fixed points in the plane is a constant that is always greater than zero. The fixed points are known as the **foci **of the hyperbola, and the line passing through the two foci is known as the transverse axis of the hyperbola.

The standard equation of the hyperbola that has the center at `(h, k)` with foci on the transverse axis is given by

`(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1`.

Where the distance between the center and each focus point is given by `c`, and `a` and `b` are the lengths of the semi-major axis and the semi-minor axis of the hyperbola, respectively.

Here, given the foci at `(-2, 5)` and `(6, 5)`, we can conclude that the center of the **hyperbola **lies on the line `y = 5`.

Also, given the transverse axis of length `4` units, we can see that the distance between the center and each of the two foci is

`c = 4 / 2

= 2`.

Thus, we have `h = 2`, `k = 5`, `c = 2`, and `a = 2`.

Therefore, the standard equation of the hyperbola is `(x - 2)^2 / 4 - (y - 5)^2 / 3 = 1`.

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Do people walk faster in an airport when they are departing (getting on a plane) or after they have arrived (getting off a plane)? An interested passenger watched a random sample of people departing and a random sample of people arriving and measured the walking speed (in feet per minute) of each. What type of study design is being performed?

Choose the correct answer below.

A. questionnaire

B. completely randomized experimental design

C. observational study

D. randomized block experimental design

### Answers

The study design being performed is an **observational study**.

The interested **passenger **watches a random sample of people who are departing (getting on a plane) and a random **sample **of people who are arriving (getting off a plane) at the airport.

The passenger measures the walking speed of each individual in terms of feet per minute. It is important to note that they are not manipulating any **variables **or assigning individuals to specific **groups**.

The study design being performed is an observational study. The passenger is simply observing and collecting data without any direct intervention or **manipulation **of variables. They are comparing the walking speeds of two separate groups (departing and arriving) but do not have control over these groups.

In an observational study, researchers gather data by observing individuals or groups and measuring variables of interest. They do not interfere with the **subjects **or manipulate variables. The goal is to understand relationships or differences that naturally occur in the observed population.

Therefore, the study design being performed is an observational study. The interested passenger is observing and measuring the walking speed of people who are **departing** and arriving at the airport without any direct **intervention **or control over the groups.

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Using the sample transaction data, you want to determine if a profit can be predicted based on customers' age and their ratings abou the product sold. What would be the null hypothesis for the population? Profit does not depend on customers' age and ratings. Profit depends on both customers' ratings and age. Profit depends on at least on customers' rating Profit depends at least on customers' age

### Answers

The null **hypothesis** for the population based on the given sample **transaction** data is that profit does not depend on customers' age and ratings.

In hypothesis testing, a null hypothesis is a statement that assumes that there is no significant difference between a set of given population **parameters**, while an alternative hypothesis is a statement that contradicts the null hypothesis and suggests that a significant difference exists. Therefore, in the given sample transaction data, the null hypothesis for the population would be: **Profit** does not depend on customers' age and ratings.However, if the alternative hypothesis is correct, it could imply that profit depends on customers' ratings and age. Therefore, the alternative hypothesis for the population could be: Profit depends on both customers' ratings and age.

Based on the null hypothesis mentioned above, a significance level or a level of significance should be set. The level of significance is the probability of rejecting the null hypothesis when it is true. The significance level is set to alpha, which is often 0.05 (5%), which means that if the test statistic value is less than or equal to the critical value, the null hypothesis should be accepted, but if the test statistic value is greater than the critical value, the null hypothesis should be rejected. After determining the null and alternative hypotheses and the level of significance, the sample data can then be analyzed using the appropriate statistical **tool** to arrive.

The null hypothesis for the population based on the given sample transaction data is that profit does not depend on customers' age and ratings.

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children's clothing company selis hand-smocked dresses for girls. The length of one particular size of dress is designed to be 28 inches, The compary regularly tests the lengths of the garments to ensure qualizy control, and if the mean length is found to be significantly longer or shorter than 28 inches, the machines must be adjusted. The most recent simple random sample of 29 dresses had a mean length of 29.15 inches with a standard deviation of 2.61 inches. Assume that the pop iation distribution is approximately normal. Perform a hypothesis test on the accuracy of the machines at the 0.10 level of significance. Step 3 of 3 : Drawa conchision and interpres the decision, Answer Keyboard shortcuts. We reject the null typothesis and conclude that there is sufficient evidence at a 0.10 invel of sgniticance that the mein length of the particular size of dress is found to be significambly ionger or shorter than 28 inches and the machines must be adjusted. We fail to reject the nuil hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance that the mean length of the particular size of dress is found to be significanty longer or shorter than 28 inches and the machines must be adjusted. We reyect the rwill hypathesis and conclude that there is irsuifficient evidence at a 0,10 leved of significance that the mean length of the particular size of dress is found to be significantly longer or shorter than 28 inches and the machines rust be adjusted. We fail to reject the null typothesis and condude that there is insuffient evidence at a 0.10 level of significance that the mean length of the particular size of dress is found to be significantly langer or shorter than 28 inches and the machines must be odjusted

### Answers

Selis, a children's clothing company, tests dress lengths for quality control. If the mean length is longer or shorter than 28 inches, machines must be adjusted. A sample of 29 dresses had a mean length of 29.15 inches with a** standard deviation** of 2.61 inches. A **hypothesis** **test** was performed at a 0.10 level, and the null hypothesis was rejected.

The children's clothing company Selis hand-smocked dresses for girls. The length of one particular size of dress is designed to be 28 inches. The company regularly tests the lengths of the garments to ensure quality control, and if the mean length is found to be significantly longer or shorter than 28 inches, the machines must be adjusted.The most recent simple random sample of 29 dresses had a mean length of 29.15 inches with a standard deviation of 2.61 inches. It is assumed that the population distribution is approximately normal.

A hypothesis test on the accuracy of the machines is performed at the 0.10 level of significance. The conclusions and interpretations of the decision are to be drawn based on the following three steps.** Null hypothesis** H0: µ = 28Alternate hypothesis H1: µ ≠ 28

Step 1: Determine the level of significance.The significance level is given as α = 0.10.

Step 2: Formulate the decision rule. Since α = 0.10, the significance level is split in half for a two-tailed test. So the critical values are -1.645 and +1.645 for a sample size of 29.

Step 3: Draw a conclusion and interpret the decision. Because the null hypothesis is µ = 28, the sample mean is 29.15, and the sample size is 29, the test statistic is calculated as follows:

z = (**sample** mean - population mean) / (standard deviation / square root of sample size)

z = (29.15 - 28) / (2.61 / sqrt(29))

z = 2.47

The p-value is P(z > 2.47) + P(z < -2.47).

The p-value for a two-tailed test is 0.013.

The test statistic is 2.47, and the critical values are -1.645 and +1.645. Since the test statistic is greater than the critical values, the null hypothesis is rejected. So, we reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance that the mean length of the particular size of dress is found to be significantly longer or shorter than 28 inches, and the machines must be adjusted. Hence, the correct option is: We reject the null hypothesis and conclude that there is sufficient evidence at a 0.10 level of significance that the mean length of the particular size of dress is found to be significantly longer or shorter than 28 inches, and the machines must be adjusted.

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Find the order of every element of (Z18, +).

### Answers

The order of every **element **in (Z18, +) is as follows:

Order 1: 0

Order 3: 6, 12

Order 6: 3, 9, 15

Order 9: 2, 4, 8, 10, 14, 16

Order 18: 1, 5, 7, 11, 13, 17

The set (Z18, +) represents the **additive group** of integers modulo 18. In this group, the order of an element refers to the smallest positive integer n such that n times the element yields the identity element (0). Let's find the order of every element in (Z18, +):

Element 0: The identity element in any group has an **order **of 1 since multiplying it by any integer will result in the identity itself. Thus, the order of 0 is 1.

Elements 1, 5, 7, 11, 13, 17: These elements have an order of 18 since multiplying them by any integer from 1 to 18 will eventually yield 0. For example, 1 * 18 ≡ 0 (mod 18).

Elements 2, 4, 8, 10, 14, 16: These elements have an order of 9. We can see that **multiplying **them by 9 will yield 0. For example, 2 * 9 ≡ 0 (mod 18).

Elements 3, 9, 15: These elements have an order of 6. Multiplying them by 6 will yield 0. For example, 3 * 6 ≡ 0 (mod 18).

Elements 6, 12: These elements have an order of 3. Multiplying them by 3 will yield 0. For example, 6 * 3 ≡ 0 (mod 18).

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Boran Stockbrokers, Inc., selects four stocks for the purpose of developing its own index of stock market behavior. Prices per share for a year 1 base period, January year 3, and March year 3 follow. Base-year quantities are set on the basis of historical volumes for the four stocks. Price per Share (s) Year 1 Stock Industry Quantity Year 1 January March Year 3 Year 3 BaseY 29.50 20.75 22.50 65.00 40.0031.00 18.00 A Oil B Computer C Steel D Real Estate 100 150 75 50 49.00 47.50 29.50 4.75 6.50 Compute the price relatives for the four stocks making up the Boran index. Round your answers to one decimal place.) Price Relative Stock March Use the weighted average of price relatives to compute the January year 3 and March year 3 Boran indexes. (Round your answers to one decimal place.)

### Answers

As per the concept of **average,** the price relatives for the four stocks making up the Boran index are as follows:

Stock A: January Year 3 - **73.88,** March Year 3 - **67.16**

Stock B: January Year 3 - **75.38, **March Year 3 - **73.08**

Stock C: January Year 3 - **82.50**, March Year 3 - **73.75**

Stock D: January Year 3 - **32.50**, March Year 3 - **18.75**

To calculate the price relatives for each stock, we need to compare the prices of each stock in different periods to the base-year price. The base-year price is the price per share in the year 1 base period. The formula for calculating the price relative is:

Price Relative = (Price in Current Period / Price in Base Year) * 100

Now let's calculate the price relatives for each **stock **based on the given data:

Stock A:

Price Relative for January Year 3 = (24.75 / 33.50) * 100 ≈ 73.88

Price Relative for March Year 3 = (22.50 / 33.50) * 100 ≈ 67.16

Stock B:

**Price **Relative for January Year 3 = (49.00 / 65.00) * 100 ≈ 75.38

Price Relative for March Year 3 = (47.50 / 65.00) * 100 ≈ 73.08

Stock C:

Price Relative for January Year 3 = (33.00 / 40.00) * 100 ≈ 82.50

Price Relative for March Year 3 = (29.50 / 40.00) * 100 ≈ 73.75

Stock D:

Price **Relative **for January Year 3 = (6.50 / 20.00) * 100 ≈ 32.50

Price Relative for March Year 3 = (3.75 / 20.00) * 100 ≈ 18.75

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Consider the following. g(x)=-9x^(2)+4x-7;h(x)=0.5x^(-2)-2x^(0.5) (a) Write the product function. f(x)=(-9x^(2)+4x-7)((0.5)/(x^(2))-2x^(0.5)) (b) Write the rate -of -change function.

### Answers

The required **rate**-of-change **function** is given as

df(x)/dx=(-9x2+4x-7)(0.5)-9x(2x-1).

a. The **product** function is given as f(x)=(-9x2+4x−7)((0.5)/(x2)−2x0.5)

Let us first simplify the second function f(x)=(0.5x−2)/x2−2√x

Now, multiply the first and second functions

f(x)=(-9x2+4x−7)(0.5x−2)/x2−2√x

Now, we get the common denominator

f(x)=(-9x2+4x−7)(0.5x−2)/(x2-2x√x+2x√x-x)

Cancelling the terms we get f(x)=(-9x2+4x−7)(0.5x−2)/(x2-x)

Factorizing the **denominator** we get f(x)=(-9x2+4x−7)(0.5x−2)/(x(x-1))

Thus, the required product function is given as f(x)=(-9x2+4x−7)(0.5x−2)/(x(x-1))

b. We know that the rate of change of a function y with respect to x is given by the derivative dy/dx.

Thus, we need to find the derivative of the function f(x) with respect to x.

Using the product rule, the **derivative** of f(x) is given as

df(x)/dx=(-9x2+4x-7)

(d/dx)(0.5x-2)+(d/dx)(-9x2+4x-7)(0.5x-2)

Differentiating the first term we get,

df(x)/dx=(-9x2+4x-7)(0.5)+(d/dx)(-9x2+4x-7)(0.5x-2)

Differentiating the second term we get,

df(x)/dx=(-9x2+4x-7)(0.5)+(-18x+4)(0.5x-2)

df(x)/dx=(-9x2+4x-7)(0.5)-9x(2x-1)

Hence, the required rate-of-change function is given as

df(x)/dx=(-9x2+4x-7)(0.5)-9x(2x-1).

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Find an equation of the plane. The plane that passes through the point (−3,1,2) and contains the line of intersection of the planes x+y−z=1 and 4x−y+5z=3

### Answers

To find an **equation** of the plane that passes through the point (-3, 1, 2) and contains the **line of intersection** of the planes x+y-z=1 and 4x-y+5z=3, we can use the following steps:

1. Find the line of intersection between the two given planes by solving the system of equations formed by equating the two plane equations.

2. Once the line of intersection is found, we can use the point (-3, 1, 2) through which the **plane** passes to determine the equation of the plane.

By solving the system of equations, we find that the line of intersection is given by the **parametric equations**:

x = -1 + t

y = 0 + t

z = 2 + t

Now, we can substitute the **coordinates** of the given point (-3, 1, 2) into the equation of the line to find the value of the **parameter** t. Substituting these values, we get:

-3 = -1 + t

1 = 0 + t

2 = 2 + t

Simplifying these equations, we find that t = -2, which means the point (-3, 1, 2) lies on the line of intersection.

Therefore, the **equation** of the plane passing through (-3, 1, 2) and containing the **line of intersection** is:

x = -1 - 2t

y = t

z = 2 + t

Alternatively, we can express the equation in the form Ax + By + Cz + D = 0 by isolating t in terms of x, y, and z from the parametric equations of the line and substituting into the plane equation. However, the resulting equation may not be as simple as the parameterized form mentioned above.

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Write Equations of a Line in Space Find the equation of the line L that passes throught point P(−5,5,3) andQ(−4,−8,−6). r(t) =+t

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To find the **equation **of the line L that passes through points P(-5, 5, 3) and Q(-4, -8, -6), we can use the point-slope form of the equation of a line in space:

r(t) = r0 + tv

where r(t) is a **vector **function that gives the position of any point on the line at time t, r0 is a known point on the line (in this case either P or Q), v is the direction vector of the line, and t is a scalar parameter.

To find v, we can take the **difference **between the two points:

v = Q - P = (-4, -8, -6) - (-5, 5, 3) = (1, -13, -9)

Now we can choose either P or Q as our known point, say P, and substitute into the equation:

r(t) = P + tv

r(t) = (-5, 5, 3) + t(1, -13, -9)

Multiplying out the **scalar **gives us:

r(t) = (-5 + t, 5 - 13t, 3 - 9t)

So the equation of the line L is:

x = -5 + t

y = 5 - 13t

z = 3 - 9t

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