Question

If, in a monopoly market, the demand for a product is p = 185 − 0.10x and the revenue function is r = px, where x is the number of units sold, what price will maximize revenue?

Answer 1

Answer:

The price that'll maximize revenue is $p=92.5$

Step-by-step explanation:

We know that the revenue function is $r = px$ where x is the number of units sold and p is the demand function given by $p = 185 - 0.10x$.

Therefore,

$r = px \\r=(185 - 0.10x)x\\r=185x-0.1x^2$

The maximums of a function are detected when the derivative is equal to zero so, to find what value of x maximizes the revenue function, we must find the derivative of the revenue function ($\frac{dr}{dx}$) and set it equal to 0.

$\frac{d}{dx} r=\frac{d}{dx} (185x-0.1x^2)\\\\\frac{d}{dx} r=\frac{d}{dx}\left(185x\right)-\frac{d}{dx}\left(0.1x^2\right)\\\\\frac{d}{dx} r=185-0.2x$

$185-0.2x=0\\185\cdot \:10-0.2x\cdot \:10=0\cdot \:10\\1850-2x=0\\1850-2x-1850=0-1850\\-2x=-1850\\\frac{-2x}{-2}=\frac{-1850}{-2}\\x=925$

Therefore, the price that'll maximize revenue is

$p = 185 - 0.10(925)\\p=92.5$