Question 1

A company’s price-demand function for memory chips is $$p(x) = 75 - 3x$$, where $$x is $$the number of millions of chips sold ($$1 \leq x \leq 20). $$What is the estimated price per chip if the company sells 11 million chips?

Accepted Answer

$$p(11) = 42$$

Question 2

Given the price-demand function $$p(x) = 75 - 3x$$, what is the company’s revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 75x - 3x^2$

Question 3

A bulldozer was purchased for $$224,000 and has a depreciated value of 100$$,000 after 8 years. The value is depreciating linearly. What is the linear equation relating value $$V to $$time $$t$$?

Accepted Answer

$$V = -15,500t + 224,000$$

Question 4

If a company manufacturing surfboards has fixed costs of $$300 per day and total costs of 5$$,100 per day at a daily output of 20 boards, what is the cost function $$C(x)$$ assuming linearity?

Accepted Answer

$$C(x) = 240x + 300$$

Question 5

What is the average cost function for the surfboard company described above?

Accepted Answer

$$C_{avg}(x) = \frac{240x + 300}{x}$$

Question 6

If $$10,000 is $$invested at a rate of $$3.95\%$$ compounded continuously for 12 years, what is the value of the investment after 12 years?

Accepted Answer

$$16,064.07$$

Question 7

A company’s total sales (in millions of dollars) $$t$$ months from now are given by $$S(t) = t + 4. $$What is $$S(4)$$ and $$S'(4)$$, and what do these values represent?

Accepted Answer

$$S(4) = 6$$, $$S'(4) = 0.25$$; 4 months from now, sales will be $$6$$ million and increasing at $$0.25$$ million per month.

Question 8

Given the price-demand equation $$p = 12,000 - 10p^2$, what is the elasticity of demand E(p$$)$$ at p = 20$?

Accepted Answer

$$E(20) = 1$$, unit elasticity

Question 9

For the function $$f(x) = x^3$ - 3x + 5$$, what are the critical numbers and the intervals where $$f(x) is $$increasing?

Accepted Answer

Critical numbers: $$x = -1$$ and $$x = 1$$; Increasing on $$(-\infty, -1)$$ and $$(1, \infty)$$

Question 10

A company manufactures and sells $$x$$ smartphones per week. The weekly price-demand and cost equations are $$p = 500 - 0.4x$$ and $$C(x) = 20,000 + 20x. $$What price should the company charge and how many phones should be produced to maximize weekly revenue?

Accepted Answer

Produce 625 phones and charge $$250 each; maximum weekly revenue is 156$$,250

Question 11

Given the cost function $$C(x) = x^3$ - 12x^2$ + 53x + 80$$, what is the cost of producing 4 bottles of sunscreen per month?

Accepted Answer

$$C(4) = 164$$

Question 12

A company’s marginal cost function for mountain bikes is $$C'(x) = 500 - x^3$ for 0$$ \leq x \leq 900$$. What is the increase in cost from producing 0 bikes to 600 bikes per month?

Accepted Answer

$$240,000$$

Question 13

Find the area bounded by $$f(x) = 2x - 2$$ and $$g(x) = x^2$ - 2.$$

Accepted Answer

$$\frac{4}{3}$$

Question 14

For the price-demand equation $$p = D(x) = 50 - 0.1x$$ and price-supply equation $$p = S(x) = 11 + 0.05x$$, what is the equilibrium point?

Accepted Answer

$$(260, 24)$$

Question 15

For the equilibrium point above, what is the consumer surplus?

Accepted Answer

$$3,380$$

Question 16

For the equilibrium point above, what is the producer surplus?

Accepted Answer

$$1,690$$

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