Question

Suppose you have 48 feet of fencing to enclose a rectangular dog pen. The function A = 24x-x2, where x = width, gives you the area of the dog pen in square feet. What width gives you the maximum area? What is the maximum area?

Answer 1

We are given

$A=24x-x^2$

where

x is width of rectangle

A is area of rectangle

Since, we have to maximize it

so, we will find it's derivative

and then we can set it to 0

and then we can solve for x

$A'=24\times 1-2x$

$A'=24-2x$

now, we can set it to 0

and then we can solve for x

$A'=24-2x=0$

$x=12$

So, width is 12  feet

Maximum area:

we can plug x=12

$A=24(12)-(12)^2$

$A=144ft^2$

So, the maximum area is

$=144ft^2$................Answer