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I. Inflation Hollingsworth Pharmaceuticals specializes in manufacturing generic medicines. Recently it developed an antibiotic with outstanding profit potential. The new antibiotic’s total costs, sales, and sales growth, as well as projected inflation, are described as follows. Total monthly costs, in dollars, to produce x units (1 unit is 100 capsules): C ( x ) = { 15 , 000 + 10 x 0 ≤ x ≤ 11 , 000 15 , 000 + 10 x + 0.001 ( x − 11 , 00 ) 2 x ≥ 11 , 000 Sales: 10,000 units per month and growing at 1.25% per month, compounded continuously Selling price: $34 per unit Inflation: Approximately 0.25% per month, compounded continuously, affecting both total costs and selling price Company owners are pleased with the sales growth but are concerned about the projected increase in variable costs when production levels exceed 11,000 units per month. The consensus is that improvements eventually can be made that will reduce costs at higher production levels, thus altering the current cost function model. To plan properly for these changes, Hollingsworth Pharmaceuticals would like you to determine when the company’s profits will begin to decrease. To help you determine this, answer the following. If you restrict your attention to total costs when x ≥ 11 , 000 , then after expanding and collecting like terms, C ( x ) can be written as follows: C ( x ) = 136 , 000 − 12 x + 0.001 x 2 for x ≥ 11 , 000 Use this form for C ( x ) with your result from Question 1(b) and with the inflationary factor e 0.0025 t to express these total costs as a function of time.

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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
Publisher: Cengage Learning
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
Publisher: Cengage Learning
ISBN: 9781305108042
Chapter 11, Problem 3EAGP1
Textbook Problem
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I. Inflation

Hollingsworth Pharmaceuticals specializes in manufacturing generic medicines. Recently it developed an antibiotic with outstanding profit potential. The new antibiotic’s total costs, sales, and sales growth, as well as projected inflation, are described as follows.

Total monthly costs, in dollars, to produce x units (1 unit is 100 capsules):

C ( x ) = { 15 , 000 + 10 x 0 x 11 , 000 15 , 000 + 10 x + 0.001 ( x 11 , 00 ) 2 x 11 , 000

Sales: 10,000 units per month and growing at 1.25% per month, compounded continuously

Selling price: $34 per unit Inflation: Approximately 0.25% per month, compounded continuously, affecting both total costs and selling price

Company owners are pleased with the sales growth but are concerned about the projected increase in variable costs when production levels exceed 11,000 units per month. The consensus is that improvements eventually can be made that will reduce costs at higher production levels, thus altering the current cost function model. To plan properly for these changes, Hollingsworth Pharmaceuticals would like you to determine when the company’s profits will begin to decrease. To help you determine this, answer the following.

If you restrict your attention to total costs when x 11 , 000 , then after expanding and collecting like terms, C ( x ) can be written as follows:

C ( x ) = 136 , 000 12 x + 0.001 x 2   for  x 11 , 000

Use this form for C ( x ) with your result from Question 1(b) and with the inflationary factor e 0.0025 t to express these total costs as a function of time.

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