1. ## Maximizing Area

I got this problem with maximizing area.

This garden is to be fenced with 120 feet of fencing. What dimensions x and y would give the garden maximum area?

I know that 3x + 2y = 120, and from that, 3x= 120 - 2y.
Other than that, I'm not sure how to tackle this one.
Any help much appreciated.

2. Originally Posted by leviathanwave
I got this problem with maximizing area.

This garden is to be fenced with 120 feet of fencing. What dimensions x and y would give the garden maximum area?

I know that 3x + 2y = 120, and from that, 3x= 120 - 2y.
Other than that, I'm not sure how to tackle this one.
Any help much appreciated.
Hello,

the area should be maximized. Therefore the main condition is:

$\displaystyle A = x \cdot y$

Now use the length of the fence as you've done above:

$\displaystyle 3x = 120-2y~\iff~x=40-\frac23 y$

Plug in this term into the equation of the area to get the characteristic function:

$\displaystyle A(y)=40y - \frac23 y^2$

The graph of this function is a parabola opening down so the maximum value is at it's vertex which you can calculate by $\displaystyle -\frac b{2a}~\implies~ -\frac{40}{-\frac43}=30$

Plug in this value into the term calculating x and you'll get x = 20.

The maximum area is 600 mē