A race track $480 m$ 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 $◻$
(Type your answer using the appropriate number of significant digits.)

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Equality occurs when $w = 120$ and $r = \frac{120}{π}$ , so the length of the side of the rectangle adjacent to the track that maximizes the area is $120$ .

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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 $2 w + 2 π r = 480$ , so $w + π r = 240$ .

Step 3 ：By AM-GM inequality, we have $240 = w + π r \geq 2 \sqrt{w π r}$ , which implies that $\sqrt{w π r} \leq 120$ .

Step 4 ：Then $w π r \leq 14400$ , so $w r \leq \frac{14400}{π}$ .

Step 5 ：The area of the rectangle is $2 w r$ , so it must satisfy $2 w r \leq \frac{28800}{π}$ .

Step 6 ：Equality occurs when $w = 120$ and $r = \frac{120}{π}$ , so the length of the side of the rectangle adjacent to the track that maximizes the area is $120$ .

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