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Printjmc
algebra senior
Problem
A rectangular field is enclosed in an athletic track, as shown below. The track consists of two edges of the field, and two semicircles. The length of the track is 400 meters. What is the largest possible area of the field, in square meters?
Solution
Let the width of the rectangle be and let the radius of each semicircle be
Then the length of the track is so By AM-GM, so Then so Then the area of the field, must satisfy Equality occurs when and so the largest possible area is
Then the length of the track is so By AM-GM, so Then so Then the area of the field, must satisfy Equality occurs when and so the largest possible area is
Final answer
\frac{20000}{\pi}