Q: A bookstore has an annual demand of $$4200$$ books. It costs $$0.40 to $$store one copy for one year, and $$125 to $$place an order. What is the optimum number of copies per order?

$$x = \sqrt{\frac{2 \times 4200 \times 125}{0.40}} \approx 1620$$

Q: The producers' surplus for the supply curve $$p = 13 + \frac{7x}{10} at$$ $$x = 300 is $$given by the area above the supply curve and below the market price. What is the producers' surplus?

$$\text{Surplus} = \$$int_0$$^{300} [B - (13 + \frac{7x}{10})] dx$$

Q: Suppose money is deposited daily into a savings account at an annual rate of $$6000. If $$the account pays $$10\%$$ interest compounded continuously, estimate the balance after $$40$$ years.

$$B = \frac{6000}{0.10}(e$$^{0.10 \times 40}$$ - 1)$$

Question 1

A recently purchased car's value is modeled by $$V(t) = 25000 e$$^{-0.20t}$$. What is the purchase price of the car?

Accepted Answer

$$25000$$

Question 2

Given $$V(t) = 25000 e$$^{-0.20t}$$, is the car appreciating or depreciating in value over time?

Accepted Answer

The car is depreciating in value.

Question 3

What is the value of the car 2 years after purchase, given $$V(t) = 25000 e$$^{-0.20t}$$?

Accepted Answer

$$V(2) = 25000 e$$^{-0.40}$$ \approx 16785$$

Question 4

The total cost of producing $$x$$ tricycles is $$C(x) = 6x^2$ + 2x + 10. $$What is the marginal cost function?

Accepted Answer

$$C'(x) = 12x + 2$$

Question 5

What is the cost of producing the 11th tricycle, given $$C(x) = 6x^2$ + 2x + 10$$?

Accepted Answer

$$C(11) - C(10) = 6(121) + 2(11) + 10 - [6(100) + 2(10) + 10] = 726 + 22 + 10 - 600 - 20 - 10 = 758 - 630 = 128$$

Question 6

The revenue function for a one-product firm is $$R(x) = 700 - \frac{1600}{x+7} - x. $$What value of $$x$$ results in maximum revenue?

Accepted Answer

$$x = 33$$

Question 7

The demand equation for beach umbrellas is $$p = \frac{1}{12}x^2$ - 10x + 300$$, $$0 \leq x \leq 60. $$What value of $$x$$ maximizes revenue?

Accepted Answer

$$x = 48$$

Question 8

A seamstress sells prom dresses for $$350$$ each if she makes $$60$$ dresses. For each dress above $$60$$, she lowers the price by $$5. $$How many dresses should she make to maximize revenue?

Accepted Answer

$$70$$

Question 9

A bookstore has an annual demand of $$4200$$ books. It costs $$0.40 to $$store one copy for one year, and $$125 to $$place an order. What is the optimum number of copies per order?

Accepted Answer

$$x = \sqrt{\frac{2 	imes 4200 	imes 125}{0.40}} \approx 1620$$

Question 10

A toy rocket is fired straight up. Its height is $$s(t) = 224t - 16t^2$. What is the rocket’s initial velocity?

Accepted Answer

$$224 ft/s$$

Question 11

A hot air balloon rises with velocity $$v(t) = 4t + 7. $$How high does it rise during the first $$10$$ seconds?

Accepted Answer

$$\$$int_0$$^{10} (4t + 7) dt = [2t^2$ + 7t]_0$$^{10}$$ = 2(100) + 70 = 200 + 70 = 270 ft$$

Question 12

A company estimates its sales grow at $$S'(t) = 15e^t. $$What is the total sales from day 2 through day 8?

Accepted Answer

$$\$$int_1$$^8 15e^t dt = 15[e^t]_1^8$ = 15(e^8$ - e^1$)$$

Question 13

A sample of radioactive material with decay constant $$0.05 is $$decaying at rate $$R(t) = -4e$$^{-0.05t}$$. How many grams decayed after the first 11$ years?

Accepted Answer

$$\$$int_0$$^{11} -4e$$^{-0.05t}$$ dt = [-80e$$^{-0.05t}$$]_0$$^{11}$$ = -80(e$$^{-0.55}$$ - 1) \approx 34.8$$ grams

Question 14

The producers' surplus for the supply curve $$p = 13 + \frac{7x}{10} at$$ $$x = 300 is $$given by the area above the supply curve and below the market price. What is the producers' surplus?

Accepted Answer

$$	ext{Surplus} = \$$int_0$$^{300} [B - (13 + \frac{7x}{10})] dx$$

Question 15

During a 12-hour period, the temperature is $$T(t) = 44 + 4t - \frac{1}{3}t^2$. What is the average temperature during this period?

Accepted Answer

$$\frac{1}{12} \$$int_0$$^{12} (44 + 4t - \frac{1}{3}t^2$) dt$$

Question 16

Suppose money is deposited daily into a savings account at an annual rate of $$6000. If $$the account pays $$10\%$$ interest compounded continuously, estimate the balance after $$40$$ years.

Accepted Answer

$$B = \frac{6000}{0.10}(e$$^{0.10 	imes 40}$$ - 1)$$

Question 17

A rectangular garden of area $$456 sq ft is to be $$surrounded on three sides by a brick wall costing $$12$$ per foot and on one side by a fence costing $$7$$ per foot. What are the dimensions that minimize the cost?

Accepted Answer

Let $$x be $$the width and $$y$$ the length. Minimize $$C = 24x + 7y$$ subject to $$xy = 456.$$

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