If a running track of 500 ft . is to be laid out enclosing a playground, the shape of which is a rectangle with a semicircle at each end, then the length of the rectangular portion such that the area of the rectangular portion is to be maximum is (in feet).

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Aicharya · Accepted Answer

Perimeter of track $=500 \mathrm{ft}$. $\Rightarrow 500=2 x+2 \pi r$

So, area of rectangular portion $=x \times 2 r=2 r(250-\pi \mathrm{r})$ $\begin{aligned} & \therefore f(r)=500 r-2 \pi r^2 \\ & f^{\prime}(r)=0 \Rightarrow 500-4 \pi r=0 \Rightarrow r=\frac{125}{\pi} \\ & \Rightarrow f^{\prime \prime}(r)=-4 \pi r \lt 0 \end{aligned}$ $\therefore$ Sides of rectangle are $2 r=\frac{250}{\pi} \mathrm{ft}$. and $x=125 \mathrm{ft}$.

If a running track of 500 ft . is to be laid out enclosing a playground, the shape of which is a rectangle…

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