Answer
$12m\times18m$, $72m$
Work Step by Step
Step 1. Assume the dimensions of the outer rectangle is $x$ meters by $y$ meters as shown in the figure.
Step 2. Given the total area as $216m^2$, we have $xy=216$ and $y=\frac{216}{x}$
Step 3. The total length of the fence will be give by $L=3x+2y=3x+\frac{432}{x}$
Step 4. Take the derivative to get $L'=3-\frac{432}{x^2}$ and an extreme will happen when $L'=0$ which gives $x=\sqrt {\frac{432}{3}}=12m$, $y=18m$, and $L=3(12)+\frac{432}{12}=72m$
Step 5. Test signs of $L'$ on both sides of $x=12$ as $..(-)..(12)..(+)..$; thus the extreme is a minimum.
Step 6. The dimensions for the outer rectangle will be $12m\times18m$ to have the smallest total length of fence. The total length of the fence needed will be $72m$ .
