Question

Two gardens. A fence of length 100 ft is to be used to enclose two

gardens. One garden is to have a circular shape, and the other to be

square.

Determine how the fence should be cut so that the sum of the areas

inside both gardens is as large as possible.

Answer 1

Let side of the square garden is a and radius of the circular garden is r.

We have to maximize $a^{2}+\pi r^{2}$

 $\begin{aligned} &\text { Subject to, } 4 a+2 \pi r=100 \\ &\text { i.e., } 2 a+\pi r=50 \end{aligned}$

The Lagrange function $L=\left(a^{2}+\pi r^{2}\right)+\lambda(2 a+\pi r-50).$

$\therefore \begin{aligned} \frac{\partial L}{\partial a} & \equiv 2 a+2 \lambda=0 \quad \Rightarrow \quad \lambda=-a \\ \frac{\partial L}{\partial r} & \equiv 2 \pi r+\lambda \pi=0 \quad \Rightarrow \lambda=-2 r . \end{aligned}$

$\begin{aligned} \therefore & 2(-\lambda)+\pi\left(-\frac{\lambda}{2}\right)=50 \\ & \Rightarrow \lambda\left(-2-\frac{\pi}{2}\right)=50 \\ & \Rightarrow \lambda=\frac{-100}{4+\pi} \\ \therefore \quad & a=\frac{100}{4+\pi} \\ \text { and, } & r=\frac{50}{4+\pi} \end{aligned}$

$\therefore$ The required fence for square $=\frac{400}{4+\pi} \mathrm{ft}$ and for circular garden $=\frac{100 \pi}{4+\pi} f_{t}.$