Afence is to be built to enclose a rectangular area of 320 square feet. the fence along three sides is to be made of material that costs 5 dollars per foot, and the material for the fourth side costs 12 dollars per foot. find the dimensions of the enclosure that is most economical to construct.

For these questions, we need to form a general equation that satisfies the conditions given.
We know:
A = 320ft²
3 fences - 5 dollars per foot and 1 fence - 12 dollars per foot.

A = 320 = xy

Eliminate one variable and solve for that:
x =   (1)

We need to now find an equation for the amount of money we are going to spend.

We can visualise it in our heads: We have two short sides to be built and two long sides to be built. One side will cost us 12 dollars, while the other three costs just 5 dollars to build.

We can now find our perimeter, which is given by 2x + 2y, but we know that x can be replaced as .

Hence, our perimeter becomes: 2() + 2y.
Now, let's write an equation for C(y), where C is our total cost.

C(y) = 5*(2* + y) + 12y) = 10* + 5y + 12y = + 17y

Now, we want to find when it's a minimum so that it will cost us the least amount of money.
Take the first derivative, and let it equal to 0.

C'(y) = -3200*y⁻² + 17 = 0
= -17
= 17

3200 = 17y²
y² =
y =

Now, to prove that it's a min. at y = , take the second derivative:

C''(y) = 3200y⁻³ and at y = , C''(y) > 0 and therefore, min. amount used.

So, at y = , it will be the least amount of money spent.

Substitute y, back into (1) to find for x.

x =

therefore, x = and y =