- 1. A beach ball is being inflated. Its volume is given by . The volume is increasing at 500 cubic centimeters per minute and the current radius is 25 cm.
- a. Give an equation for .
- b. Give the corresponding rate of change of the radius (to the nearest hundredth)? Answer with complete sentence(s) and include units.
- 2. A shed will have square ends and a sloped roof. The distance from the top of the square to the top peak is half the width of the square. The rectangular volume is 640 cubic feet. (This does not include any space in the triangular region). The cost of siding for all rectangular sides is $.75 per square foot. Nicer siding will be used in the triangular region. It will cost $1.50 per square foot. Shingles for roof will cost $1 per square foot.
- a. Clearly define any variables used
- b. Give an equation for volume of rectangular area.
- c. Give the equation for the Surface area.
- d. Give the equation for cost of siding and roofing the shed.
- e. Give the equation for the derivative of cost.
- f. Find the value of the derivative when the width of the square is 10 ft. Explain the meaning in terms of the problem.
- g. Find when the derivative is equal to zero. Explain the significance of this point.
- h. Give the dimensions of the shed that will minimize the cost of the shed.
- i. Give the minimum cost of siding and roofing.
- 3. A large tire company plans to sell 1 million tires during the next year. Sales tend to be roughly the same throughout the year. Setting up each production run costs the company $20,000. Carrying costs based on the average number of tires in storage amount to $5 for one tire for one year.
- a. Find the cost if there were 5 production runs during the year.
- b. give the derivative of the cost
- c. Give the number of production runs and the size of each run that will minimize the overall cost of producing the tires.
- 4. Joe can sell 600 replacement windows throughout the year for $150 each. For each $10 dollar increase in price by he will be able to sell 25 less windows.
- a. Find an equation for revenue where x is the number of $10 increases. [Hint: Price multiplied by quantity gives revenue.]
- b. Give the derivative of the revenue
- c. Find the price that will optimize his weekly revenue and give the optimal revenue.
- d. If he made $50 per window before the price change and any price change comes from or adds to his profit directly how much money will he make at the new price?

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Mar 30th, 2014
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