A fence must be built to enclose a rectangular area of 5000 ftsquared. fencing material costs $ 3 per foot for the two sides facing north and south and $6 per foot for the other two sides. find the cost of the least expensive fence.

They've given us the area of the enclosure: 20000 ft^2

So if I were to write an equation for area I would get:

A = xy (Length * Width)

or

20000 = xy

Since we're trying to optimize the cost of the fence, we need to write an equation that represents this, and find its derivative. To do this, we'll first have to find the perimeter of the enclosure.

P = 2x + 2y

Now let's use the information given about the cost of each side to write an equation for the cost.

C = 3 (2x) + 6 (2y)

C = 6x + 12y

Now let's use the area equation to get rid of one of the variables in the cost equation:

20000 = xy

Solve for y:

y = 20000/x

Now let's substitute this into our cost equation:

C = 6x + 12y

C = 6x + 12 (20000/x)

C = 6x + 240000/x

Take the derivative:

C' = 6 - 240000/x^2

Whenever you're doing optimization problems, always set you're derivative to 0.

0 = 6 - 240000/x^2

Now solve for x by multiplying through by x^2.

0 = 6x^2 - 240000

Solve for x:

x = 200

Since we're trying to find the least cost, we substitute this value into our cost equation to get our answer:

C = 6 (200) + 240000/200

C = 2400