A running track of $440 \mathrm{ft}$ is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then the lengths of its side are

• A

  $70 \mathrm{ft}$ and $110 \mathrm{ft}$
• B

  $80 \mathrm{ft}$ and $120 \mathrm{ft}$
• Correct answer
  C

  $35 \mathrm{ft}$ and $110 \mathrm{ft}$
• D

  $35 \mathrm{ft}$ and $120 \mathrm{ft}$

Solution

Perimeter $=440 \mathrm{ft}$ $\Rightarrow \quad 2 x+\pi r+\pi r=440$ $2 x+2 \pi r=440$ $$ \begin{aligned} A & =\text { Area of the rectangular portion }=x 2 r \\ A & =x \frac{(440-2 x)}{\pi}=\frac{1}{\pi}\left(440 x-2 x^2\right) \end{aligned} $$ Let $\frac{d A}{d x}=\frac{1}{\pi}(440-4 x)=0$ $$ \begin{aligned} & \Rightarrow x=110 \text { for which } \frac{d^2 A}{d x^2} < 0 \\ & \Rightarrow A \text { is maximum when } x=110 \\ & \Rightarrow 2 r=\frac{440-2 x}{\pi}=\frac{440-220}{22 / 7}=70 \\ & \therefore \quad r=35 \mathrm{ft} \text { and } x=110 \mathrm{ft} \end{aligned} $$

Hence, the correct answer is option: 'C'.