Maxima Minima problem about a fencing

Question

A rectangle field of area $2400 m^2$ is to be fenced off along a straight road. The front fencing costs $P80$ per meter, that of the sides and back costs $P40$ per meter. Determine the minimum total cost required.

I couldn't think of a proper equation for this problem. Appreciate the help!

Answer 1

HINT:

We have $$\text{Length of Side}=\frac{\text{Area}}{\text{Length of Front Fence}}$$ and $$\begin{align}\text{Cost}&=80\times\text{Length of Front Fence}+40\times 2\times\text{Length of Side}+80\times\text{Length of Back Fence}\\&=120\times\text{Length of Front Fence}+80\times\frac{\text{Area}}{\text{Length of Front Fence}}\end{align}$$

Can you now find an expression for cost in mathematical notation and differentiate it with respect to the length of the front fence?

Answer 2

let $x$ be the length of the front of the rectangle, and $y$ be the length of the side.

$x\cdot y = 2400$

Total cost $T = 80(x)+ 40(x+2y)$

$T = 80x+40x+80\cdot \dfrac{2400}{x}$

You need to minimize $T$

I Assume you can do the rest.