Question 1

A company manufactures and sells $$x$$ smartphones per week. The weekly price-demand equation is $$p = 500 - 0.5x. $$What is the revenue function $$R(x) in $$terms of $$x$$?

Accepted Answer

$$R(x) = (500 - 0.5x)x$$

Question 2

Given the price-demand equation $$p = 500 - 0.5x$$ for smartphones, what value of $$x$$ maximizes the weekly revenue?

Accepted Answer

$$x = 250$$

Question 3

For the smartphone company, what price should be charged per phone to maximize weekly revenue?

Accepted Answer

Question 4

What is the maximum weekly revenue the smartphone company can achieve?

Accepted Answer

$$R = 62,500$$

Question 5

A company manufactures and sells $$x$$ digital cameras per week. The weekly price-demand equation is $$p = 40 - 0.8x. $$What is the revenue function $$R(x) in $$terms of $$x$$?

Accepted Answer

$$R(x) = (40 - 0.8x)x$$

Question 6

For the digital camera company, what value of $$x$$ maximizes the weekly revenue?

Accepted Answer

$$x = 25$$

Question 7

What price should the digital camera company charge per camera to maximize weekly revenue?

Accepted Answer

$$p = 20$$

Question 8

What is the maximum weekly revenue the digital camera company can achieve?

Accepted Answer

$$R = 500$$

Question 9

Given the cost function for smartphones $$c(x) = 20,000 + 135x$$, what is the profit function $$P(x) in $$terms of $$x$$?

Accepted Answer

$$P(x) = (500 - 0.5x)x - (20,000 + 135x)$$

Question 10

Suppose the smartphone company wants to maximize profit instead of revenue. Which of the following steps is correct for finding the number of phones to produce for maximum profit?

Accepted Answer

Set the derivative of the profit function $$P(x) to $$zero and solve for $$x$$

Question 11

If the digital camera company produces 10 cameras per week, what is the weekly cost using $$c(x) = 20 + 16x$$?

Accepted Answer

$$c(10) = 180$$

Question 12

For the smartphone company, if 100 phones are produced and sold, what is the price per phone according to the price-demand equation $$p = 500 - 0.5x$$?

Accepted Answer

$$p = 450$$

Question 13

If the digital camera company sells 30 cameras per week, what is the price per camera according to $$p = 40 - 0.8x$$?

Accepted Answer

$$p = 16$$

Question 14

Which of the following best describes the economic meaning of the vertex of the revenue function $$R(x)$$ for these companies?

Accepted Answer

It represents the production level that maximizes total revenue.

Question 15

If the smartphone company increases production beyond the revenue-maximizing quantity, what will happen to the total revenue?

Accepted Answer

Total revenue will decrease.

Question 16

For the digital camera company, what is the marginal revenue function $$MR(x)$$?

Accepted Answer

$$MR(x) = 40 - 1.6x$$

Question 17

If the smartphone company wants to break even, which equation should they solve?

Accepted Answer

$$(500 - 0.5x)x = 20,000 + 135x$$

Question 18

Suppose the digital camera company wants to maximize profit. Which of the following is the correct profit function?

Accepted Answer

$$P(x) = (40 - 0.8x)x - (20 + 16x)$$

Question 19

If the smartphone company produces 0 phones, what is the weekly cost?

Accepted Answer

$$c(0) = 20,000$$

Question 20

For the digital camera company, what is the fixed cost?

Accepted Answer

$$20$$

Business Calculus: Maximizing Revenue with Price-Demand Functions

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