Q: A company’s revenue function is $$R(x) = 60x$$, and its cost function is $$C(x) = 1200 + 30x. $$What is the profit when 50 units are sold?

$$P(50) = 60 \times 50 - (1200 + 30 \times 50) = 3000 - (1200 + 1500) = 300$$

Q: A company’s revenue function is $$R(x) = 25x$$ and its cost function is $$C(x) = 10x + 300. $$What is the profit when 40 units are sold?

$$P(40) = 25 \times 40 - (10 \times 40 + 300) = 1000 - (400 + 300) = 300$$

Q: A company’s cost function is $$C(x) = 1000 + 12x$$ and its revenue function is $$R(x) = 20x. If $$the company sells 200 units, what is the profit?

$$P(200) = 20 \times 200 - (1000 + 12 \times 200) = 4000 - (1000 + 2400) = 600$$

Q: A company’s cost function is $$C(x) = 700 + 18x$$ and its revenue function is $$R(x) = 25x. $$What is the profit when 100 units are sold?

$$P(100) = 25 \times 100 - (700 + 18 \times 100) = 2500 - (700 + 1800) = 0$$

Question 1

Suppose a company’s cost function is given by $$C(x) = 500 + 20x$$, and its revenue function is $$R(x) = 50x. $$What is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 50x - (500 + 20x)$$

Question 2

A company manufactures widgets. The cost function is $$C(x) = 2000 + 15x$$, and the revenue function is $$R(x) = 35x. $$What is the break-even point (number of widgets $$x$$ where profit is zero)?

Accepted Answer

$$x = 100$$

Question 3

If the cost function for producing $$x$$ units is $$C(x) = 1000 + 8x$$, what is the marginal cost?

Accepted Answer

$$8$$

Question 4

A company’s revenue function is $$R(x) = 60x$$, and its cost function is $$C(x) = 1200 + 30x. $$What is the profit when 50 units are sold?

Accepted Answer

$$P(50) = 60 	imes 50 - (1200 + 30 	imes 50) = 3000 - (1200 + 1500) = 300$$

Question 5

Which of the following best describes the marginal cost?

Accepted Answer

The additional cost incurred by producing one more unit of output.

Question 6

A company’s equipment is purchased for $$\50$$,000$ and is expected to have no salvage value after 10 years. Using linear depreciation, what is the annual depreciation expense?

Accepted Answer

$$\5$$,000$

Question 7

If the cost function is $$C(x) = 100 + 12x$$ and the revenue function is $$R(x) = 20x$$, what is the equilibrium point?

Accepted Answer

$$x = 12.5$$

Question 8

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$What is the maximum profit, and at how many units is it achieved?

Accepted Answer

Maximum profit is $$300 at$$ $$x = 10$$ units.

Question 9

A business has a cost function $$C(x) = 500 + 25x$$ and a revenue function $$R(x) = 40x. If $$the company wants to make a profit of at least $$\1$$,000$, what is the minimum number of units it must sell?

Accepted Answer

$$100$$

Question 10

A machine is purchased for $$\24$$,000$$ and depreciates linearly to $$\$$0$$ over 6 years. What is its book value after 4 years?

Accepted Answer

$$\8$$,000$

Question 11

If the marginal cost is constant at $$15$$, and the fixed cost is $$600$$, what is the total cost function $$C(x)$$?

Accepted Answer

$$C(x) = 600 + 15x$$

Question 12

A company’s revenue function is $$R(x) = 25x$$ and its cost function is $$C(x) = 10x + 300. $$What is the profit when 40 units are sold?

Accepted Answer

$$P(40) = 25 	imes 40 - (10 	imes 40 + 300) = 1000 - (400 + 300) = 300$$

Question 13

Which of the following statements about the break-even point is correct?

Accepted Answer

It is the point where total revenue equals total cost.

Question 14

A company’s cost function is $$C(x) = 300 + 18x$$ and its revenue function is $$R(x) = 30x. $$What is the profit function?

Accepted Answer

$$P(x) = 30x - (300 + 18x)$$

Question 15

A company’s cost function is $$C(x) = 1000 + 12x$$ and its revenue function is $$R(x) = 20x. If $$the company sells 200 units, what is the profit?

Accepted Answer

$$P(200) = 20 	imes 200 - (1000 + 12 	imes 200) = 4000 - (1000 + 2400) = 600$$

Question 16

A company’s equipment is purchased for $$\36$$,000$$ and is expected to have a salvage value of $$\$$6,000$$ after 10 years. Using linear depreciation, what is the annual depreciation expense?

Accepted Answer

$$\3$$,000$

Question 17

If the cost function is $$C(x) = 400 + 10x$$ and the revenue function is $$R(x) = 25x$$, what is the minimum number of units that must be sold to make a profit?

Accepted Answer

$$x = 27$$

Question 18

A company’s profit function is $$P(x) = 12x - 500. $$What is the break-even quantity?

Accepted Answer

$$x = 41.67$$

Question 19

A company’s cost function is $$C(x) = 700 + 18x$$ and its revenue function is $$R(x) = 25x. $$What is the profit when 100 units are sold?

Accepted Answer

$$P(100) = 25 	imes 100 - (700 + 18 	imes 100) = 2500 - (700 + 1800) = 0$$

Business Calculus Applications: Profit, Cost, Revenue, and Marginal Analysis

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