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Applications of derivatives: maximizing area and revenue

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4. A Norman window consists of a rectangle with a semi-circle mounted on top (see the figure). What are the dimensions of the Norman window with the largest area and a fixed perimeter of P meters?

5. A bus company will charter a bus that holds 50 people to groups of 35 or more. If a group contains exactly 35 people, each person pays $60. In larger groups, everyone's fare is reduced by $1 for each person in excess of 35.

(a) What is the revenue, if the bus is chartered to 35 people? 36 people?
37 people?
(b) Find a formula for the revenue in terms of the number of people
chartering the bus.
(c) Determine the size of the group(s) for which the bus company's
revenue will be the greatest.
(Note: you need to check that your answer makes physical sense and
that it is correct as the physical constraints allow.)

See attached file for full problem description.

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Solution Summary

It composed of two applications of derivative problems: maximizing the area with a fixed perimeter and maximzing the revenue of a bus company. The solution is well presented and detailed.

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Pfizer and Mylan

1. The income statement and balance sheet for Galaxy Interiors (for 2011) are shown below:

A. What is the firm's operating cash flow for 2011 (as defined in Chapter 2)?
B. What is the "Net cash from operating activities" on the Statement of Cash Flows (see Chapter 3)?
C. What was the amount of dividends paid in 2011?
D. What is the amount of the net capital spending for 2011?

2. ABC Pharmaceuticals has current assets of $6,500, net fixed assets of $37,500, current liabilities of $4,900, and long term debt of $16,800. In addition, the firm just received approval from the FDA to start selling product XYZ, which it has developed (from R&D spending) over the past several years. This product (which is owned by ABC) has a market value of $750,000 (this is the present value of the expected future cash flows).

A. What is the amount of Shareholder's Equity on this firm's accounting balance sheet?
B. Do you think the firm's asset market value is greater than, less than, or equal to its asset book value? How do you know? (You do not need to give a precise numerical answer to this question.)

3. Given the following information for Cutting-Edge Cure Company, calculate depreciation expense: sales = $34,000; administrative costs (excluding depreciation) = 12,000; R&D costs (excluding depreciation) = 4,000; addition to retained earnings = $4,300; dividends paid = $1,200; interest expense = $2,300; tax rate = 35%.

4. What is the sustainable growth rate (assuming the following ratios are constant)?

1/1.20 = .833
1/[1/1.20) - 1]
1.200 - 1 = D/E .200

(.08) (1.46)(1 +.200)
ROE = .14016

1 - .32 = .68

[.14016 (.68)] / (1-.14016 (.68)
= 0.953/.9047
SGR = .1053

5. Natural Country Cures has sales of $35 million, total debt of $36 million, and a debt-to-equity ratio of 1.2. The firm's net profit margin is 6 percent.

A. What is the firm's net income (in dollars)?
B. What is the firm's ROA?
- Net Income/Assets =(NI/Sales) x (Sales/Assets) =
C. What is the firm's ROE?
- Net Income/Equity = (Net Income/Assests) x (Assests/Equity) = ROA x (Assets/Equity)

Question 6 is on the next page

6. Define the gross profit margin (GPM) as follows:

Note that cost of goods sold (COGS) includes only the raw material (or inventory) acquisition costs, the direct production costs, and the packaging costs of items sold.

In addition, define the R&D/Sales ratio as:

Obtain the 2011 annual reports for Pfizer and Mylan Inc. from the internet. You must use numbers obtained from the firms' 2011 income statements in parts A, C, and D (show your work!).

A. Calculate the gross profit margin for each firm for 2011.
B. Why do these firms have dramatically different gross profit margins? What fundamental difference in their business models causes this difference? (Use your own words).
C. Calculate the R&D/Sales ratio for each firm in 2011.
D. Why do these firms have dramatically different R&D/Sales ratios? (This answer is similar to part B, but specifically address R&D expense in Part D).

Question 7 is on the next page

7. The most recent financial statements for Watchtower Inc. are shown here (assume no income taxes, and ignore interest expense):

Income Statement

Sales $5,100
Costs 3,480
Net Income 1,620

Balance Sheet

Assets 14,500 Debt 10,200
Equity 4,300
Total 14,500 Total 14,500

Assets and costs are proportional to sales; debt and equity are not. No dividends are paid. Next year's sales are projected to be $5,967.

A. What is the external financing needed?
B. What maximum sales amount (in dollars) could the firm support next with no external financing?
C. What is the sales growth rate implied by the sales amount you identify in part B? What is this growth rate called? (Hint: This term is in the book!)

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