Answer to Question #276101 in Calculus for Ash

Question #276101

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.

1
Expert's answer
2021-12-07T06:39:34-0500

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

We have to maximize a2+πr2a^{2}+\pi r^{2}

  Subject to, 4a+2πr=100 i.e., 2a+πr=50\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=(a2+πr2)+λ(2a+πr50).L=\left(a^{2}+\pi r^{2}\right)+\lambda(2 a+\pi r-50).

La2a+2λ=0λ=aLr2πr+λπ=0λ=2r.\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}

2(λ)+π(λ2)=50λ(2π2)=50λ=1004+πa=1004+π and, r=504+π\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 =4004+πft=\frac{400}{4+\pi} \mathrm{ft} and for circular garden =100π4+πft.=\frac{100 \pi}{4+\pi} f_{t}.



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