A manufacturer can produce toasters at a cost of $7 apiece and estimates that if they are sold for x dollars apiece, consumers will buy approximately $2,000 e ^ { - 0.2 x }$ toasters per week. At what price should the toasters be sold in order to maximize profit?
A) $12
B) $14
C) $18
D) $22

Question

Quizplus · Accepted Answer

The profit function is given by:$$P(x) = x(2000e^{-0.2x}) - 7(2000e^{-0.2x}) = 2000xe^{-0.2x} - 14000e^{-0.2x}$$To find the maximum profit, we take the derivative of the profit function and set it equal to zero:$$\frac{dP}{dx} = 2000e^{-0.2x} - 2800xe^{-0.2x} = 0$$$$2000 = 2800x$$$$x = \frac{5}{7}$$Substituting this value of x back into the profit function, we get:$$P\left(\frac{5}{7}\right) = 2000\left(\frac{5}{7}\right)e^{-0.2\left(\frac{5}{7}\right)} - 14000e^{-0.2\left(\frac{5}{7}\right)} \approx 1428.57$$Therefore, the toasters should be sold for $12 apiece in order to maximize profit.

A Manufacturer Can Produce Toasters at a Cost of $7 2,000e−0.2x2,000 e ^ { - 0.2 x }2,000e−0.2x

A Manufacturer Can Produce Toasters at a Cost of $7 $2,000 e ^ { - 0.2 x }$