A builder wishes to fence in 72000 m2 of land in a rectangular shape. For security reasons, the fence along the front part of the land will cost $23 per meter, while the fence for the other three sides will cost $13 per meter. How much of each type of fence should the builder buy to minimize the cost of the fence? What is the minimum cost? Use 4 decimal places for the fence amounts and round to 4 decimal places.

a) the fence along the front and the back side should have 228.0350 m

each and the sides should have a length of 315.7410 m

b) the minimum cost is $ 16418.526

Explanation:

denoting 1 as the 23$/m fence and 2 as the 13$/m fence, "a" as the length of the front fence and "b" of the 3 other 3 sides, the cost function will be

C = C1 * a + C2 (2b+a)

resticted to a*b = A (Area=72000m2)

therefore b=A/a

replacing in the cost function

C=C1*a+C2 (2A/a + a)

if we derive with respect to a, the minimum will be at dC/da=0. therefore

dC/da = C1 + C2 (-2A/a² + 1) = 0

solving for a

a = √[2*A / (1+C1/C2) ]

replacing values

a = √[2*72000m2 / (1+23$/m/13$/m) ] = 228.0350 m

thus b will be

b = A/a = 72000 m2/228.0350 m=315.7410 m

and the minimum cost will be

C = C1 * a + C2 (2b+a) = 23$/m * 228.0350m + 13$/m * (2*315.7410 m + 228.0350m) = $ 16418.526