Mathematico | Mathematico

A farmer is building a pen. The farmer uses $120 m$ of fencing and builds a pen in the following shape:

Farmer pen

Find the value of $A a b$ , where $A$ is the minimum possible area of the pen, and $a, b$ are the dimensions for which this minimum occurs.

Hint

The perimeter of the shape is given by the expression

4 a + 2 b

Solution

Considering the perimeter, we have

\begin{aligned} 4 a + 2 b & = 120 \\ \Rightarrow & b & = 60 - 2 a \end{aligned}

Substituting this into the area, we get

\begin{aligned} a^{2} + b^{2} & = a^{2} + (60 - 2 a)^{2} \\ = a^{2} + 3600 - 240 a + 4 a^{2} \\ = 5 a^{2} - 240 a + 3600 \\ = 5 [a^{2} - 48 a] + 3600 \\ = 5 [{(a - 24)}^{2} - 576] + 3600 \\ = 5 {(a - 24)}^{2} + 72 \end{aligned}

The minimum area for the pen is $720$ and it occurs when $a = 24$ and $b = 12$ .