Question

PLEASEE can someone help me with this and show the work

Answer 1

Answer:

Width to give maximum area is 12 feet. Maximum area is 144 square feet.

Step-by-step explanation:

Given:

Width is given as $x$.

Area of the rectangular pen is given as, $A=24x-x^{2}$

For maximum area, the derivative of area with respect to $x$ must be 0. So,

$\frac{dA}{\mathrm{d} x}=\frac{\mathrm{d} }{\mathrm{d} x}(24x-x^{2})\\0=24-2x\\2x=24\\x=\frac{24}{2}=12$

Therefore, for maximum area, the width should be 12 feet.

Now, plug in 12 ft for $x$ in the expression for area to calculate maximum area. This gives,

Maximum area, $A_{m}$, is given as:

$A_{m}=24x-x^{2}\\A_{m}=24(12)-12^{2}=288-144=144\textrm{ }ft^{2}$

Therefore, the maximum area is 144 square feet.