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) = 500x - 0.5x^2$

Question 2

For the smartphone company, what value of $$x$$ maximizes the weekly revenue, given $$p = 500 - 0.5x$$?

Accepted Answer

$$x = 250$$

Question 3

Given the smartphone price-demand equation $$p = 500 - 0.5x$$, what price should the company charge to maximize weekly revenue?

Accepted Answer

$$p = 375$$

Question 4

What is the maximum weekly revenue for the smartphone company, using the price-demand equation $$p = 500 - 0.5x$$?

Accepted Answer

$$R_{max} = 62,500$$

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 $$R(x)$$?

Accepted Answer

$$R(x) = 40x - 0.8x^2$

Question 6

For the digital camera company, what value of $$x$$ maximizes weekly revenue, given $$p = 40 - 0.8x$$?

Accepted Answer

$$x = 25$$

Question 7

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

Accepted Answer

$$p = 20$$

Question 8

What is the maximum weekly revenue for the digital camera company, using the price-demand equation $$p = 40 - 0.8x$$?

Accepted Answer

$$R_{max} = 500$$

Question 9

For the smartphone company, the cost function is $$c(x) = 20,000 + 135x. $$What is the profit function $$P(x)$$?

Accepted Answer

$$P(x) = 500x - 0.5x^2$ - 20,000 - 135x$$

Question 10

Suppose the smartphone company wants to maximize profit instead of revenue. Which equation should be solved to find the optimal $$x$$?

Accepted Answer

$$\frac{d}{dx}P(x) = 0$$

Question 11

If the digital camera company produces 25 cameras per week, what is the weekly cost using $$c(x) = 20 + 16x$$?

Accepted Answer

$$c(25) = 420$$

Question 12

For the smartphone company, if 250 phones are produced and sold, what is the weekly cost using $$c(x) = 20,000 + 135x$$?

Accepted Answer

$$c(250) = 53,750$$

Question 13

For the digital camera company, what is the profit when 25 cameras are produced and sold at the revenue-maximizing price?

Accepted Answer

$$P(25) = 500 - 420 = 80$$

Question 14

If the smartphone company produces 250 phones, what is the profit at the revenue-maximizing price?

Accepted Answer

$$P(250) = 62,500 - 53,750 = 8,750$$

Question 15

Which of the following best describes the relationship between price and demand in the smartphone company scenario?

Accepted Answer

As price decreases, demand increases.

Question 16

If the digital camera company wants to maximize profit, which function should they optimize?

Accepted Answer

Profit function $$P(x)$$

Question 17

For the smartphone company, what is the marginal revenue when $$x = 250$$?

Accepted Answer

Question 18

If the digital camera company increases production from 25 to 26 units, what is the approximate increase in revenue?

Accepted Answer

$$\Delta R \approx R(26) - R(25)$$

Business Calculus: Maximizing Revenue with Price-Demand Functions

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