Answer:
Width to give maximum area is 12 feet. Maximum area is 144 square feet.
Step-by-step explanation:
Given:
Width is given as [tex]x[/tex].
Area of the rectangular pen is given as, [tex]A=24x-x^{2}[/tex]
For maximum area, the derivative of area with respect to [tex]x[/tex] must be 0. So,
[tex]\frac{dA}{\mathrm{d} x}=\frac{\mathrm{d} }{\mathrm{d} x}(24x-x^{2})\\0=24-2x\\2x=24\\x=\frac{24}{2}=12[/tex]
Therefore, for maximum area, the width should be 12 feet.
Now, plug in 12 ft for [tex]x[/tex] in the expression for area to calculate maximum area. This gives,
Maximum area, [tex]A_{m}[/tex], is given as:
[tex]A_{m}=24x-x^{2}\\A_{m}=24(12)-12^{2}=288-144=144\textrm{ }ft^{2}[/tex]
Therefore, the maximum area is 144 square feet.
