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):
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 inflation is assumed to be compounded continuously, the selling price and total costs must be multiplied by the factor . In addition, if sales growth is assumed to be compounded continuously, then sales must be multiplied by a factor of the form , where r is the monthly sales growth rate (expressed as a decimal) and t is time in months. Use these factors to write each of the following as a function of time t:
selling price p per unit (including inflation).
number of units x sold per month (including sales growth).
total revenue. (Recall that .)
Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!Get Started