Question

Let's say I have $480 to fence in a rectangular garden. The fencing for the north and south sides of the garden costs $10 per foot and the fencing for the east and west sides costs $15 per foot. How can I find the dimensions of the largest possible garden.?

Answer 1

Let's call the length of the N and S sides `x` (feet) and the other two we will call `y` (also in feet)

Then the cost of the fence will be:

`2*x*$10` for N+S and `2*y*$15` for E+W

Then the equation for the total cost of the fence will be:

`20x+30y=480`

We separate out the `y`:
`30y=480-20x->y=16-2/3 x`

Area:
`A=x*y`, replacing the `y` in the equation we get:
`A=x*(16-2/3 x)=16x-2/3 x^2`

To find the maximum, we have to differentiate this function, and then set the derivative to `0`

`A'=16-2*2/3x=16-4/3 x=0`

Which solves for `x=12`
Substituting in the earlier equation `y=16-2/3 x=8`

Answer:
N and S sides are 12 feet
E and W sides are 8 feet
Area is 96 square feet