A manufacturer can produce radios at a cost of $10 apiece and estimates that if they are sold for x dollars apiece, consumers will buy approximately $200 e ^ { - 0.2 x }$ radios per month. The price at which the manufacturer should sell the radios to maximize the profit is
A) $10
B) $15
C) $18
D) $20

Question

Quizplus · Accepted Answer

To maximize profit, we need to find the price $x$ that maximizes the profit function. Profit is given by $P(x) = (x - 10) \cdot 200e^{-0.2x}$, where $x$ is the selling price per radio and $10$ is the cost to produce each radio. To find the maximum, we differentiate $P(x)$ with respect to $x$ and set the derivative equal to zero. Differentiating $P(x)$ gives: $\frac{dP}{dx} = 200e^{-0.2x} - 40x e^{-0.2x}$. Setting this equal to zero and solving for $x$ will give the price that maximizes profit. Simplifying the equation, we find that $x = 15$ is the value that maximizes profit, which corresponds to choice B.

A Manufacturer Can Produce Radios at a Cost of $10 200e−0.2x200 e ^ { - 0.2 x }200e−0.2x

A Manufacturer Can Produce Radios at a Cost of $10 $200 e ^ { - 0.2 x }$