To minimize the cost, the builder should but 200 meters of fence costs at $2 per meter and 800 meters of fence costs at $1 per meter. The minimum cost is $1,200.
Recall that if we have a function f(x), then at the extremum point, its derivative is equal to zero.
f ' (x) = 0
In the given problem, let:
p = length of the rectangle
q = width of the rectangle
Then,
p x q = 60,000 or q = 60,000/p
Assume that the front side is p, then the function that describe the cost is:
f(p,q) = 2xp + 1 x (p + q + q)
f(p,q) = 3p + 2q
f(p) = 3p + 2 x 60,000/p
f(p) = 3p + 120,000/p
Take the derivative:
f '(p) = 3 - 120,000/p² = 0
p² = 40,000
p = 200 meters
Substitute p = 200 to get q,
q = 60,000/200 = 300
Hence, the type of fences the builder have to buy to minimize the cost is:
200 meters fence with cost $2 per meter and 200+300+300= 800 meters fence with cost $1 per meter.
The minimum cost is:
f(p) = 3p + 120,000/p
f(min) = f(200) = 3x200 + 120,000/200 = $1,200
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