calculus area question

Question

A fence is to be built to enclose a rectangular area of 230 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 16 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

Answer 1

Set up a system of equations:

Let L = length (both sides are $3 per foot)

Let W = width (one side is $3 per foot and the other side is $16 per foot)

A = L * W = 230 → use known area to solve for W in terms of L → W = 230 / L

cost = $3 (2L + W) + $16 (W) = 6L + 3W + 16W = 6L + 19W

substitute W = 230 / L into cost equation so there is only one variable:

cost = f(L) = 6L + 19W = 6L + 19*(230/L) = 6L + (4370 / L)

For optimization problems, find the derivative of the cost equation, set equal to 0, and solve for L:

f'(L) = 6 - (4370 / L²) = 0

6 = 4370 / L²

L² = 4370 / 6

L = √(4370 / 6) = 26.988 ft

Use L to solve for W:

W = 230 / L = 230 / 26.988 = 8.522 ft

The dimensions of the enclosure are 26.988 ft by 8.522 ft