A race track $480 \mathrm{~m}$ long is built around an area that is a rectangle with a semicircle at each end. Find the length of the side of the rectangle adjacent to the track if the area of the rectangle is to be a maximum.
The length of the side of the rectangle adjacent to the track that maximizes the area is $\square$
(Type your answer using the appropriate number of significant digits.)
Answer
Equality occurs when \(w = 120\) and \(r = \frac{120}{\pi}\), so the length of the side of the rectangle adjacent to the track that maximizes the area is \(\boxed{120}\).
Step 1 ：Denote the width of the rectangle as \(w\) and the radius of each semicircle as \(r\).
Step 2 ：The length of the track is \(2w + 2 \pi r = 480\), so \(w + \pi r = 240\).
Step 3 ：By AM-GM inequality, we have \(240 = w + \pi r \ge 2 \sqrt{w \pi r}\), which implies that \(\sqrt{w \pi r} \le 120\).
Step 4 ：Then \(w \pi r \le 14400\), so \(wr \le \frac{14400}{\pi}\).
Step 5 ：The area of the rectangle is \(2wr\), so it must satisfy \(2wr \le \frac{28800}{\pi}\).
Step 6 ：Equality occurs when \(w = 120\) and \(r = \frac{120}{\pi}\), so the length of the side of the rectangle adjacent to the track that maximizes the area is \(\boxed{120}\).
