Donut Delights Inc has determined that when x donuts are mad

Donut Delights. Inc. has determined that when x donuts are made daily, the profit P. in dollars, is given by P(x) = - 0.001 x^2 + 3.2x-1200 What is the company\'s profit if 600 donuts are made daily? How many donuts should be made daily in order to maximize the company\'s profit?

Solution

The profit P(x) = - 0.001x² + 3.2x – 1200 where x is the number of donuts made daily.

(a) If 600 donuts are made daily, then x = 600, and on substituting this value of x in the equation of the prfit function, we have P(600) = - 0.001( 600)² + 3.2 (600) – 1200 = - 360 + 1920 – 1200 = 360 . Thus the Company’s profit is $ 360 if 600 donuts are made daily.

(b) For maximum profit, dP/ dx = 0 i.e., - 0.001(2x) + 3.2 = 0 or, - 0.002x + 3.2 = 0 or, x = 3.2/ 0.002 = 1600. We can verify this result, by substituting x = 1600, 1500 and 1700 (say) in the equation of P(x). We have P(1600) = - 0.001(1600)² + 3.2 (1600) – 1200 = - 2560 + 5120 – 1200 = $ 1360. Similarly, P(1500) = $ 1350 and P (1700) = $ 1350.Thus, the maximum profit is $ 1360.