Problem
a. A rectangular pen is built with one side against a barn. If $100 \mathrm{~m}$ of fencing are used for the other three sides of the pen, what dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of $225 \mathrm{~m}^{2}$. What are the dimensions of each pen that minimize the amount of fence that must be used? a. To maximize the area of the pen, the sides perpendicular to the barn should be $25 \mathrm{~m}$ long and the side parallel to the barn should be $50 \mathrm{~m}$ long. (Type exact answers, using radicals as needed) b. To minimize the amount of fence that must be used, each of the sides perpendicular to the barn should be $\mathrm{m}$ long and each of the sides parallel to the barn should be $\square \mathrm{m}$ long. (Type exact answers, using radicals as needed)
Solution
Step 1 ：Let the sides perpendicular to the barn be of length \(x\). Notice that there are a total of \(100\) meters of fencing. Therefore, the side parallel to the barn has length \(100-2x\).
Step 2 ：The area to be maximized is \(100x-2x^2\).
Step 3 ：Completing the square results in \(-2(x-25)^2+1250\), which is maximized when \(x=25\).
Step 4 ：Therefore, the side parallel to the barn has length \(100-2(25)=\boxed{50}\) meters.
Step 5 ：The sides perpendicular to the barn should be \(\boxed{25}\) meters long.
