Question

You need to enclose an area of 300 square feet with fence. If the area is rectangular, what is the least amount of fence you need?

Answer 1

Given

An enclose an area of 300 square feet with fence. If the area is rectangular, what is the least amount of fence

Solution

Then perimeter

$Perimeter=2(x+\frac{300}{x})$$\begin{gathered} \frac{dp}{dx}=2(1-\frac{300}{x^2})=0 \\ \\ 1-\frac{300}{x^2}=0 \\ \\ 1=\frac{300}{x^2} \\ \\ x^2=300 \\ x=\sqrt{300} \end{gathered}$

Now substitute x into the perimeter

$\begin{gathered} Perimeter=2(\sqrt{300}+\frac{300}{\sqrt{300}}) \\ perimeter\text{ =amount} \\ Amount=2(\sqrt{300}+\frac{300}{\sqrt{300}}) \\ Amount=40\sqrt{3} \\ Amount=69.28203 \end{gathered}$

The least amount of fence 69.28203ft