Benjamin used regression analysis to fit quadratic relations to monthly revenue, R, and total cost, TC, data with the following results, where Q is quantity.
R = ?0.008Q2 + 32Q
TC = 0.005Q2 + 2.2Q + 10
(a) Using a spreadsheet, plot R and TC. Estimate the quantities QBE and Qp, where the maximum profit should occur. Estimate the amount of profit at this quantity.
(b) The profit relation P = R ? TC and calculus can be used to determine the quantity Qp at which the maximum profit will occur and the amount of this profit. The equations are:
Profit = aQ2 + bQ + c
Qp = ?b/2a
Maximum profit = ?b2/4a + c
Use these relations to confirm your graphical estimate of QP. (Your instructor may ask you to derive the relations above.)
(a) Plot shows maximum quantity at approximately 1000 units. Profit estimate is $16,790 per month. Breakeven points QBE are at approximately 0 and 2300 units.
(b) Profit = R – TC = (-.008-.005) Q² + (32-2.2)Q - 10
= -.013Q² + 29.8Q - 10
Qp = -b/2a = -29.8/2(-.013)
= 1146 units
Profit = -b²/4a + c = -29.8² / 4(-.013) - 10
= $17,068
If Q = 1146 is entered into the spreadsheet column A, the profit is displayed as $17,068
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