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.

Question

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.)

Answer 1

(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