Respuesta :
Answer:
The price that'll maximize revenue is [tex]p=92.5[/tex]
Step-by-step explanation:
We know that the revenue function is [tex]r = px[/tex] where x is the number of units sold and p is the demand function given by [tex]p = 185 - 0.10x[/tex].
Therefore,
[tex]r = px \\r=(185 - 0.10x)x\\r=185x-0.1x^2[/tex]
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 ([tex]\frac{dr}{dx}[/tex]) and set it equal to 0.
[tex]\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[/tex]
[tex]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[/tex]
Therefore, the price that'll maximize revenue is
[tex]p = 185 - 0.10(925)\\p=92.5[/tex]
