What happens if you can write a function as a composite in different​ ways? Do you get the same derivative each​ time? The Chain Rule says you should. Find
dy/dx if y=x by using the Chain Rule with y as a composite of the following functions. Complete parts​ (a) and​ (b) below.

Answer:

2x² + 5x + c = 0

For this quadratic equation to have one double root, the discriminant must equal 0.

5² - 4(2)(c) = 0

25 - 8c = 0

c = 25/8

2x² + 5x is not a perfect square because the coefficient of x², 2, is not a perfect square.

2x² + 5x is not a perfect square.

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Scientific notation is a way of expressing numbers that are very large or very small in a concise and standardized format. It consists of two parts: a coefficient and a power of 10.

To write a number in scientific notation, we express it as a product of a number between 1 and 10 (coefficient) and a power of 10. The coefficient should be greater than or equal to 1 and less than 10.

Let's convert the given hair diameters into scientific notation:

1. Palest blond hair diameter: 0.0017 centimeters

To convert it to scientific notation, we can express 0.0017 as 1.7 multiplied by a power of 10. We count the number of decimal places needed to shift the decimal point until we have a coefficient between 1 and 10.

0.0017 = 1.7 × 0.001

Since we moved the decimal point three places to the right, the power of 10 is -3.

Therefore, the palest blond hair diameter in scientific notation is:

0.0017 centimeters = 1.7 × 10^(-3) centimeters

2. Black hair diameter: 0.018 centimeters

Similarly, we can express 0.018 as 1.8 multiplied by a power of 10. This time, we only need to shift the decimal point one place to the right.

0.018 = 1.8 × 0.01

Since we moved the decimal point two places to the right, the power of 10 is -2.

Therefore, the black hair diameter in scientific notation is:

0.018 centimeters = 1.8 × 10^(-2) centimeters

In summary:

The palest blond hair diameter is written in scientific notation as 1.7 × 10^(-3) centimeters.

The black hair diameter is written in scientific notation as 1.8 × 10^(-2) centimeters.

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Given:

Annual salary = $71,000

Total deductions = $17,555.48

Number of installments = 12

Annual salary after deductions:

[tex]\displaystyle \text{Annual salary after deductions} = \text{Annual salary} - \text{Total deductions}[/tex]

[tex]\displaystyle \text{Annual salary after deductions} = \$71,000 - \$17,555.48[/tex]

[tex]\displaystyle \text{Annual salary after deductions} = \$53,444.52[/tex]

Monthly paycheck:

[tex]\displaystyle \text{Monthly paycheck} = \frac{{\text{Annual salary after deductions}}}{{\text{Number of installments}}}[/tex]

[tex]\displaystyle \text{Monthly paycheck} = \frac{{\$53,444.52}}{{12}}[/tex]

[tex]\displaystyle \text{Monthly paycheck} \approx \$4,453.71[/tex]

Therefore, John will receive approximately $4,453.71 with each monthly paycheck.

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♥️ [tex]\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}[/tex]

The measure of the angle is 65 degrees

From the question, we have the following parameters that can be used in our computation:

The circle

From the circle, we have

R = 90

Also, we have

P = 1/2 * 50

P = 25

Using the above as a guide, we have the following:

PQR = 180 - 90 - 25

Evaluate

PQR = 65

Hence, the measure of the angle is 65 degrees

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The square miles of the park the students have cleaned before lunch is ⅜ square miles

Area of the park to be cleaned = 1/2 area of square mile.

Fraction of park the class had cleaned before lunch break = 3/4

Number of square miles they had cleaned before lunch = Fraction of park the class had cleaned before lunch break × Area of the park to be cleaned

= 3/4 × 1/2

= (3 × 1) / (4 × 2)

= 3 / 8 square miles

Hence, the students had cleaned ⅜ square miles before lunch break.

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After 15 years, you would have approximately $3,040.19 in the account.

To calculate the future value of the account after 15 years, we can use the formula for compound interest:

[tex]A = P(1 + r/n)^{(nt)[/tex]

Where:

A = the future value of the account

P = the initial deposit or principal amount

r = the annual interest rate (expressed as a decimal)

n = the number of times interest is compounded per year

t = the number of years

In this case, the initial deposit or principal amount (P) is $2000, the annual interest rate (r) is 3% or 0.03, the interest is compounded annually (n = 1), and the time period (t) is 15 years.

Plugging in the values into the formula, we have:

[tex]A = 2000(1 + 0.03/1)^{(1\times15)[/tex]

Simplifying the calculation:

[tex]A = 2000(1.03)^{15[/tex]

Using a calculator or computer software, we can evaluate this expression:

[tex]A \approx 2000(1.03)^{15}[/tex]

≈ 2000(1.52009438088)

≈ $3,040.19.

Therefore, after 15 years, you would have approximately $3,040.19 in the account.

This calculation assumes that you make the $2000 deposit each year without any changes or interruptions and that the interest rate remains constant at 3% compounded annually.

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The volume of the cylinder is 56.52 cubic units

From the question, we have the following parameters that can be used in our computation:

Using the above as a guide, we have the following:

V = πr²h

Substitute the known values in the above equation, so, we have the following representation

V = 3.14 * 3² * 2

Evaluate

V = 56.52

Hence, the volume is 56.52 cubic units

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Mr. Smith has $17 left to spend on each student.

An equation is a mathematical statement that is formed when two algebraic expressions are equated using an equal sign.

The total number of students Mr. Smith has 17

The money he can spent on each student is $22

The total money he can spend is $22 · 17

Total cost of calculators is $83.64

Let x is the amount left to spend on each student.

The equation that can be formed is

[tex]17 \times \text{x} = 22\times17 - 83.64[/tex]

[tex]\text{x} = 22 - \huge \text(\dfrac{83.64}{17}\huge \text)[/tex]

[tex]\text{x} = 22 - 5[/tex]

[tex]\bold{x = \$17}[/tex]

Therefore, Mr. Smith has $17 left to spend on each student.

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Mr. Smith has 17 students in his class.

He can spend $22 on each student to buy math supplies for the year.

The calculators cost a total of $83.64.

The total amount he has to spend on all 17 students is $22 x 17 = $374.

The amount he spent on calculators was $83.64.

So the amount he has left to spend is $374 - $83.64 = $290.36

He has $290.36 left to spend on 17 students.

Therefore, the amount he has left to spend on each student is $290.36 divided by 17 students, which is $17.07.

So the answer is: After buying the calculators, Mr. Smith has $17.07 left to spend on each student.

In summary:

$22 - Amount he can spend per student

$83.64 - Total cost of calculators

$374 - Total amount he has to spend on all 17 students ($22 x 17)

$290.36 - Amount he has left to spend after buying calculators ($374 - $83.64)

$17.07 - Amount he has left to spend on each student ($290.36 / 17 students)

Answer:

2(l+b)

2(75+22)

=194

Step-by-step explanation:

the perimeter of the pool is calculated with a formula.

2(l+b)

The domain of f(x) is x > -4, so the range of f⁻¹(x) is y > -4.

The range of f(x) is y ≥ 5, so the domain of the f⁻¹(x) is x ≥ -4.

In Mathematics and Geometry, an inverse function simply refers to a type of function that is obtained by reversing the mathematical operation in a given function (f(x)).

In this exercise, you are required to determine the inverse of the function f(x). This ultimately implies that, we would have to swap (interchange) both the independent value (x-value) and dependent value (y-value) as follows;

f(x) = y = √(x - 5) - 4

x = √(y - 5) - 4

√(y - 5) = x + 4

By taking the square of both sides of the function, we have:

y - 5 = (x + 4)²

f⁻¹(x) = (x + 4)² + 5

Therefore, the domain and range of f(x) and f⁻¹(x) are given by:

domain of f(x) = x > -4

range of f(x) = y ≥ 5

domain of f⁻¹(x) = x ≥ -4.

range of f⁻¹(x) = y > -4.

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The maximum price you can offer for a home is approximately $82,352.94 for a 17% down payment.

To determine the maximum price you can offer for a home, you need to calculate 17% of the total price, which will be your down payment of $14,000.

Let's assume the maximum price you can offer for the home is P dollars.

According to the given information, the down payment requirement is 17% of the total price. So, we can set up the following equation:

[tex]0.17P = $14,000[/tex]

To solve for P, divide both sides of the equation by 0.17:

P = $14,000 / 0.17

Calculating this expression, we find:

P ≈ $82,352.94

Therefore, the maximum price you can offer for a home in order to have enough money for the 17% down payment is approximately $82,352.94.

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According to the information, it can be inferred that Leah reaches a maximum distance of 8 blocks away from her house. Additionally, the journey takes 23 minutes.

To graph the information for Leah's commute, we must take into account the distance (in blocks) and the time it takes her (in minutes) to travel to the bank and then return home.

In this case we must take into account the speed (blocks per minute) that Leah has because this modifies the time and distance that she will take in total. According to the above, the graph ascends, then remains stable for 2 minutes, then rises more steeply and remains stable for 7 minutes, finally descends to 0 steadily.

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Answer:

In the given options, the parameter is represented by the letter "p." Therefore, the correct answer is option d.

Step-by-step explanation:

12x5/3 simplifies to the whole number 20.

To simplify the expression 12x5/3, we need to perform the multiplication and division operations.

First, let's multiply 12 and 5:

12 * 5 = 60

Now, let's divide the result by 3:

60 / 3 = 20

Therefore, 12x5/3 simplifies to the whole number 20.

However, since you mentioned "In m2" in your question, it seems that there may be a variable involved. If the expression 12x5/3 represents an area in square meters, then we need more information about the variable x to provide a more specific answer.

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Answer:     (10 1/2) Months or 10.5 Months

Hence, in Ten and a half (10 1/2) Months, The Company will recover its investment in Program X.

Step-by-step explanation:

Divide the savings in each month. (Twelve Months (12) in a year) by yearly savings to the company (C) from Praogram X that is equal to $40,000

        40000 / 12   =   10000 / 3

                              =     3333.33333

        35000 / 40000  =  .875

                                     =   7/8

                                     =  0.875

      0 .875  *   12            =   10 1/2  or  10.5 Months  

Hence, In 10 1/2  or  10.5 Months, The Company will recover its investment in Program X.

I hope this helps you!

Answer:

A. Through a given point not on a given line, there is exactly one line parallel to the given line.

Step-by-step explanation:

The fifth postulate of Euclidean geometry is often referred to as the Parallel Postulate.

A. Through a given point not on a given line, there is exactly one line parallel to the given line. This statement is actually the Parallel Postulate, so it is the correct answer.

B. All right angles are equal to one another. This statement is actually the Fourth Postulate, also known as the Right Angle Postulate.

C. Any straight line segment can be extended indefinitely. This statement is the Second Postulate, also known as the Line Extension Postulate.

D. A circle can be drawn with any center and radius. This statement is the Third Postulate, also known as the Circle Postulate.

Therefore, the correct answer is:

A. Through a given point not on a given line, there is exactly one line parallel to the given line.

Answer:

Since there are 10 boys and 6 of them equal 400Kg and the average of each boy is 60kg the remaining 4 boys should equal around 250kgs

Answer:

Step-by-step explanation:

The standard deviation is a measure of how spread out the data.

In this case, the standard deviation is relatively small, which means that the data is tightly clustered around the mean.

This suggests that the running times of the first seven runners were fairly evenly distributed, with no extreme outliers.

Formula to calculate the standard deviation manually:

[tex]\boxed{\boxed{ \tt Standard \:deviation =\sqrt{\frac{\sum (data\: point - mean)^2}{number \:of \:data \:points}}}}[/tex]

we have:

For First seven:

n=7

First of all, we need to find mean:

[tex]\tt Mean=\frac{ \sum{x}}{n}=\frac{9.63 + 9.75 + 9.79 + 9.8 + 9.88 + 9.94}{ 7 }= 9.824 seconds[/tex]

In this case, the standard deviation would be calculated as follows:

[tex]\tt Standard\: deviation = \sqrt{\frac{(9.63 - 9.82)^2 + (9.75 - 9.82)^2 + (9.79 - 9.82)^2 + (9.8 - 9.82)^2 + (9.88 - 9.82)^2 + (9.94 - 9.82)^2}{7}}\\\tt Standard\: deviation =\sqrt{ \frac{0.0859}{7}}\\\tt Standard\: deviation = \sqrt{0.01227}\\\tt Standard\: deviation = 0.1107[/tex]

             This confirms that the data for the first seven runners in the men's 100 meters final at the London Olympics can be fit into a normal distribution.

             The standard deviation of 0.11 seconds means that the data is tightly clustered around the mean. This suggests that the running times of the first seven runners were fairly evenly distributed, with no extreme outliers.

Therefore, the answer to your questions are:

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By printing 12,632 copies, the printer accounts for the expected spoilage rate of 5%, ensuring that she will have at least 12,000 clean copies available for the job.

To ensure that the printer has 12,000 clean copies after accounting for a spoilage rate, she needs to calculate the total number of copies she should print.

Let's denote the total number of copies she should print as x.

The spoilage rate is given as 5%, which can be expressed as a decimal as 0.05. So, the number of good copies she expects is 100% - 5% = 95% of the total number of copies.

Therefore, the equation to find the number of copies she should print is:

0.95 * x = 12,000

To solve for x, we divide both sides of the equation by 0.95:

x = 12,000 / 0.95

Using a calculator, we find:

x ≈ 12,631.58

Since we cannot print a fraction of a copy, the printer should round up to the nearest whole number. She should print 12,632 copies to ensure 12,000 clean copies.

By printing 12,632 copies, the printer accounts for the expected spoilage rate of 5%, ensuring that she will have at least 12,000 clean copies available for the job.

It's important to consider the spoilage rate and print a sufficient number of extra copies to compensate for any potential issues during the printing process and meet the required clean copy quantity.

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The correct statement regarding the translation in this problem is given as follows:

The graph is translated down 1 unit.

A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

The functions for this problem are given as follows:

Due to the subtraction by 1, the function was translated down 1 unit.

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Answer:

y = -8/3x + 8

Step-by-step explanation:

Step 1:  Identify which values we have and need to find in the slope-intercept form:

The general equation of the slope-intercept form of a line is given by:

y = mx + b, where

Since we're told that the y-intercept is 8, this is our b value in the slope-intercept form.

Step 2:  Find m, the slope of the line:

Thus, we can find m, the slope of the line by plugging in (3, 0) for (x, y) and 8 for b:

0 = m(3) + 8

0 = 3m + 8

-8 = 3m

-8/3 = m

Thus, the slope is -8/3.

Therefore, the the equation of the line in slope-intercept form whose x-intercept is 3 and whose y-intercept is 8 is y = -8/3x + 8.

Optional Step 3:  Check the validity of the answer:

Thus, we can check that we've found the correct equation in slope-intercept form by plugging in (3, 0) and (0, 8) for (x, y), -8/3 for m, and 8 for b and seeing if we get the same answer on both sides of the equation when simplifying:

Plugging in (3, 0) for (x, y) along with -8/3 for m and 8 for b:

0 = -8/3(3) + 8

0 = -24/3 + 8

0 = -8 = 8

0 = 0

Plugging in (0, 8) for (x, y) along with -8/3 for m and 8 for b;

8 = -8/3(0) + 8

8 = 0 + 8

8 = 8

Thus, the equation we've found is correct as it contains the points (3, 0) and (0, 8), which are the x and y intercepts.

Answer:

15,659,568

Step-by-step explanation:

To calculate the expression 2308208 * 12 - 12038928, we can follow the order of operations (PEMDAS/BODMAS):

1. Multiply: 2308208 * 12 = 27698496

2. Subtract: 27698496 - 12038928 = 15659568

Therefore, 2308208 * 12 - 12038928 equals 15,659,568.

Algebraically, it is shown that y and [tex]y^{(-1)}[/tex]  are inverse functions.

Part A:

To determine if the given relation [tex]y = 2(x - 5)^2 - 6[/tex]  is a function, we need to verify that for every input value of x, there is a unique output value of y.

In this case, it is indeed a function because for each x-value, there is only one corresponding y-value.

To determine if it is one-to-one, we need to check if different input values yield different output values.

For the given relation, it is not one-to-one because if we substitute two different x-values that yield the same output, the function fails the one-to-one criterion.

In this case, any two different x-values that are equidistant from 5 (such as 4 and 6) will yield the same output.

To make the relation one-to-one, we can restrict the domain. One possible restriction on the domain is to limit x to values less than 5 or greater than 5.

By excluding the x-values that are equal to 5, we can ensure that each input x corresponds to a unique output y, satisfying the one-to-one condition.

Part B:

To determine y^(-1) (the inverse of y), we need to swap the roles of x and y and solve for y.

Let's start with the equation y = 2(x - 5)^2 - 6:

Swap x and y: x = 2(y - 5)^2 - 6.

Rearrange the equation to solve for y:

x + 6 = 2(y - 5)^2.

Divide both sides by 2:

(x + 6) / 2 = (y - 5)^2.

Take the square root of both sides:

±√[(x + 6) / 2] = y - 5.

Add 5 to both sides:

y = ±√[(x + 6) / 2] + 5.

Hence, y^(-1) = ±√[(x + 6) / 2] + 5.

Part C:

To prove that y and y^(-1) are inverse functions, we need to show that when the two functions are composed, they yield the original input.

Let's start with y = 2(x - 5)^2 - 6.

Substitute y^(-1) into the equation:

y = 2(√[(x + 6) / 2] + 5)^2 - 6.

Simplify the equation:

y = 2[(x + 6) / 2] + 10√[(x + 6) / 2] + 25 - 6.

Cancel out terms and simplify further:

y = x + 6 + 10√[(x + 6) / 2] + 19.

Since we obtain x as the result when we compose y and y^(-1), it proves that y and y^(-1) are indeed inverse functions.

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The quadratic equation shown in the graph is:

y = (x + 1)² + 5

A quadratic equation with a leading coefficient "a" and a vertex (h, k) is written as:

y = a*(x - h)² + k

Here the vertex is at (-1, 5), then we can write:

y = a*(x + 1)² + 5

The function also passes through  (-4, 14), then:

14 = a*(-4 + 1)² + 5

14 = a*9 + 5

14 - 5 = 9a

9 = 9a

9/9 = a

1 = a

The quadratic equation is:

y = (x + 1)² + 5

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The measure of the indicated arc is 134 degrees

from the question, we have the following parameters that can be used in our computation:

The circle

The measure of the indicated arc is equal to the angle subtended at the center

using the above as a guide, we have the following:

Angle = 134 degrees

This means that

Arc = 134 degrees

Hence, the measure of the indicated arc is 134 degrees

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Answer: Please see attached images

(2nd image is the answer, 1st image is it worked out)

Step-by-step explanation:

You are given "Use synthetic division to divide [tex]x^3 + 6x^2 +15x + 24[/tex] by [tex]x + 3[/tex]"

1) You first take the coefficients from the expression, [tex]x^3 + 6x^2 +15x + 24[/tex], and place them in a row.

    |      1      6     15     24

    |__________________

2) Next you take the expression, x + 3, and take the last coefficient and put it to the left.

3   |      1      6     15     24

    |__________________

3) Next you drop the 1 (this is the case for the first number of any synthetic division problem)

3   |      1      6     15     24

    |__ ↓_______________

           1

4) Now you multiply 3 and 1, then place that product under 6.

3   |      1      6     15     24

    |__ ↓__  3___________

           1

5) Then you subtract the column: 6 - 3, and put the answer below

3   |      1     6     15     24

    |__ ↓__  3___________

           1      3

6) Now you repeat this process (steps 4-5) but with the following numbers. So repeating step 4, You multiply 3 and 3, and put that product under 15.

3   |      1     6     15     24

    |__ ↓__  3__ 9________

           1      3

7) Repeating step 5, you subtract the column: 15 - 9, and put the answer below.

3   |      1     6     15     24

    |__ ↓__  3__ 9________

           1      3      6

8) Once again, you repeat steps 4-5. So repeating step 4, you multiply, 3 and 6, and you place the product under 24.

3   |      1     6     15     24

    |__ ↓__  3__ 9___18____

           1      3      6

9) Repeat step 5, you subtract the column: 24 - 18, and put the answer below.

3   |      1     6     15     24

    |__ ↓__  3__ 9___18____

           1      3      6      | 6  

To find the quotient, that is result of the synthetic division but with its variables,  and not including the remainder.

Quotient: [tex]x^2+3x+6[/tex]

The remainder is the remaining number that is at the bottom of the last column.

Remainder: 6

Answer:

  see attached

Step-by-step explanation:

You want the squares of the numbers 2 to 11.

The calculator output attached shows the squares of the numbers 2 through 11.

__

Additional comment

If your knowledge of multiplication tables extends through 10×10, then there is only one more you need to memorize here. It is actually a good idea to learn the squares of numbers through 20², and be able to identify the cubes of numbers through 10³. (12³ = 1728 is also useful for cubic inches in a cubic foot.)

<95141404393>

Answer:

Step-by-step explanation:

A is the answer

Calculate the area of 3 rectangles

4x10=40 x 3 rectangles = 120

calculate the area of 2 triangles

A=1/2bh, since you have triangles, no need to divide by 2

base x height

4 x 3.5 = 14

Surface Area is 120 + 14 = 134

When x = 25 and dx = 7 units per day, the rate of change of time (dt) with respect to x is 1 / (7 units/day).

In the given scenario, we have two functions: R(x) and C(x). Let's use the given information to calculate the rate of change of x with respect to time (dt) when x = 25 and dx = 7 units per day.

To find the rate of change, we can differentiate the functions with respect to x. Let's differentiate both R(x) and C(x) to obtain their derivatives:

dR/dx = 2

dC/dx = 0.02x + 0.3

Now, we need to find dt, the rate of change of time with respect to x. To do this, we can use the chain rule of differentiation, which states that dt/dx = 1 / (dx/dt)

Given that dx = 7 units per day, we can calculate dx/dt:

dx/dt = 7 units/day

Now, we can find dt by taking the reciprocal:

dt/dx = 1 / (7 units/day)

Therefore, the rate of change of time with respect to x is 1 / (7 units/day).

Note that the units cancel out in the reciprocal operation, leaving us with units/day.

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The variances for August are as follows:

1. Materials Price Variance: $332,000 (U)

2. Materials Quantity Variance: $40,275 (U)

3. Labor Rate Variance: $5,750 (U)

4. Labor Efficiency Variance: $0 (None)

5. Variable Overhead Rate Variance: $1,900 (U)

6. Variable Overhead Efficiency Variance: $0 (None)

Using  the following formula:

Materials Price Variance = (Actual Price - Standard Price) * Actual Quantity

Calculating the variances:

1. Materials Price Variance:

Actual Price = $56,000

Standard Price = $23.10 per yard

Actual Quantity = 10,000 yards

Materials Price Variance = ($56,000 - $23.10 * 10,000) = $332,000 (U)

2. Materials Quantity Variance:

Actual Quantity = 10,000 yards

Standard Quantity = 2,500 sets * 3.3 yards per set = 8,250 yards

Standard Price = $23.10 per yard

Materials Quantity Variance = (10,000 - 8,250) * $23.10 = $40,275 (U)

3. Labor Rate Variance:

Actual Rate = $9,250

Standard Rate = $3.50 per hour

Actual Hours = 1,000 hours

Labor Rate Variance = ($9,250 - $3.50 * 1,000) = $5,750 (U)

4. Labor Efficiency Variance:

Actual Hours = 1,000 hours

Standard Hours = 2,500 sets * 0.40 hours per set = 1,000 hours

Standard Rate = $3.50 per hour

Labor Efficiency Variance = (1,000 - 1,000) * $3.50 = $0 (None)

5. Variable Overhead Rate Variance:

Actual Rate = $4,500

Standard Rate = $1.60 per hour

Actual Hours = 1,000 hours

Variable Overhead Rate Variance = ($4,500 - $1.60 * 1,000) = $1,900 (U)

6. Variable Overhead Efficiency Variance:

Actual Hours = 1,000 hours

Standard Hours = 2,500 sets * 0.40 hours per set = 1,000 hours

Standard Rate = $1.60 per hour

Variable Overhead Efficiency Variance = (1,000 - 1,000) * $1.60 = $0 (None)

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