Hello, falcarius!

This is a trick question . . . certainly a *tricky* question.

A 400-km racetrack is to be built with two straight sides and semicircles at the ends.

Find the dimensions of the track that encloses the maximum area.

Given: The two long sides of the rectangle __>__ 100 m.

The diameter of the 2 semicircles __>__20 m.

I went through the expected procedure for an optimization. Code:

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L

Let $\displaystyle L$ = length of the straight tracks.

Let $\displaystyle r$ = radius of the semicircular tracks.

The total length of the straight tracks is: $\displaystyle 2L$.

The total length of the curved tracks is: $\displaystyle 2\pi r$

. . Hence, we have: .$\displaystyle 2L + 2\pi r \:=\:400\quad\Rightarrow\quad L \:=\:200 - \pi r$ . **[1]**

The area of the rectangle is: $\displaystyle 2rL$.

The area of the two semicircles is: $\displaystyle \pi r^2$.

. . Hence, the total area is: .$\displaystyle A \;=\;\pi r^2+ 2rL$ . **[2]**

Substitute [1] into [2]: .$\displaystyle A \;=\;\pi r^2+ 2r(200 - \pi r)\quad\Rightarrow\quad A \;=\;400r - \pi r^2$

Differentiate and equate to zero: .$\displaystyle A' \;=\;400 - 2\pi r \;=\;0$

. . Hence: .$\displaystyle r \,=\,\frac{200}{\pi}$

Substitute into [1] and we get: .$\displaystyle L \,=\,0$ . . . . What's going on?

It is fairly well-known that, for a given perimeter,

. . the figure with maximum area is a __circle__.

So for maximum area, the track should be circular.

But we are told that the straight tracks must be at least 100 meters (each).

. . What do we do now?

It can be shown that, as $\displaystyle L$ increases, the area decreases.

. . Hence, for maximum area, we will use the __least__ $\displaystyle L$, 100.

Under the restrictions, maximum area is attained when $\displaystyle L = 100,\;r = \frac{100}{\pi}$