Q: Given the cost function $$C(x) = 20x + 10,000$$, what is the total cost of producing 100 suits?

$$C(100) = 20 \times 100 + 10,000 = 12,000$$

Q: A company’s total cost function is $$C(x) = 20x + 10,000. If $$the company produces 400 suits, what is the total cost?

$$C(400) = 20 \times 400 + 10,000 = 18,000$$

Q: A company’s cost function is $$C(x) = 20x + 10,000. If $$the company increases production from 100 to 200 suits, by how much does the total cost increase?

$$C(200) - C(100) = (20 \times 200 + 10,000) - (20 \times 100 + 10,000) = 4,000$$

Question 1

Given the points $$(-2, 6)$$ and $$(-4, 9)$$, what is the slope $$m of $$the line passing through them?

Accepted Answer

$$m = \frac{6 - 9}{-2 - (-4)} = \frac{-3}{2} = -1.5$$

Question 2

Which of the following best describes a linear function?

Accepted Answer

A function of the form $$f(x) = mx + b$$, where $$m$$ and $$b$$ are constants.

Question 3

Find the equation of the line with slope $$m = 3$$ that passes through the point $$(-1, -5).$$

Accepted Answer

Question 4

Given the cost function $$C(x) = 20x + 10,000$$, what is the total cost of producing 100 suits?

Accepted Answer

$$C(100) = 20 	imes 100 + 10,000 = 12,000$$

Question 5

For the cost function $$C(x) = 20x + 10,000$$, what is the fixed cost?

Accepted Answer

$$10,000$$

Question 6

If the total cost function for producing $$x$$ units is $$C(x) = 20x + 10,000$$, what is the variable cost per unit?

Accepted Answer

$$20$$

Question 7

Find the equation of the line passing through the points $$(-1, 4)$$ and $$(3, -6). $$What is its $$y-$$intercept?

Accepted Answer

Slope $$m = \frac{-6 - 4}{3 - (-1)} = \frac{-10}{4} = -2.5. $$Equation: $$y = -2.5x + 1.5$$. $$y-$$intercept: $$(0, 1.5)$$

Question 8

A company’s total cost function is $$C(x) = 20x + 10,000. If $$the company produces 400 suits, what is the total cost?

Accepted Answer

$$C(400) = 20 	imes 400 + 10,000 = 18,000$$

Question 9

Which of the following statements about fixed and variable costs is correct?

Accepted Answer

Fixed costs do not change with the number of units produced, while variable costs increase as more units are produced.

Question 10

Given the equation of a line in point-slope form $$y - y_1$ = m(x - x_1$)$$, what is the equation of the line with slope $$m = -2$$ passing through $$(3, 7)$$?

Accepted Answer

$$y - 7 = -2(x - 3)$$

Question 11

A clothing company’s cost function is $$C(x) = 20x + 10,000. If $$the company wants to keep total costs below $$15,000$$, what is the maximum number of suits they can produce?

Accepted Answer

$$x \u003c 250$$

Question 12

Which of the following is the correct interpretation of the $$y-$$intercept in the cost function $$C(x) = 20x + 10,000$$?

Accepted Answer

The $$y-$$intercept represents the fixed costs, which are $$10,000$$ dollars.

Question 13

If a company’s cost function is $$C(x) = 20x + 10,000$$ and it produces no suits ($$x = 0$$), what is the total cost?

Accepted Answer

$$10,000$$

Question 14

Which of the following is the correct general form of a linear function?

Accepted Answer

$$f(x) = mx + b$$

Question 15

A company’s cost function is $$C(x) = 20x + 10,000. If $$the company increases production from 100 to 200 suits, by how much does the total cost increase?

Accepted Answer

$$C(200) - C(100) = (20 	imes 200 + 10,000) - (20 	imes 100 + 10,000) = 4,000$$

Question 16

Given the equation $$y = mx + b$$, what does the parameter $$m$$ represent in the context of a business cost function?

Accepted Answer

The variable cost per unit produced.

Question 17

If the equation of a line is $$y = -2x + 5$$, what is the slope and $$y-$$intercept?

Accepted Answer

Slope: $$-2$$, $$y-$$intercept: $$5$$

Question 18

A company’s cost function is $$C(x) = 20x + 10,000. If $$the company produces 0 suits, what portion of the cost is due to variable costs?

Accepted Answer

$$0$$

Question 19

Which of the following is NOT a property of a linear function?

Accepted Answer

Its graph is a parabola.

Business Calculus Study Guide: Linear Functions, Slope, and Cost Applications

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