A rectangular corral with a total area of 60 square meters is to be fenced off a

Question

A rectangular corral with a total area of 60 square meters is to be fenced off and then divided into 2 sections with a fence down the middle. The fencing for the outside costs $9 per meter, while the dividing fence costs $12 per meter. What dimensions will minimize the cost?

a. the object is to minimize cost. the objective function is C= ????

b. you must have 60 square meters of corrral. the constraint equation is: ?????

c. The dimensions that will minimize the cost are x= ??? and y= ???

Please show me the steps of your answer. thanks!

Explanation / Answer

A = Lw
60 = Lw
60/L = w

The cost of the outside fencing: 9(2L + 2w) = 18(L + w) = 18(L + 60/L)
The cost of the dividing fence: 12w = 12(60/L) = 720/L

The total cost is:

C(L) = 18(L + 60/L) + 720/L
C(L) = 18[(L + 60/L) + 40/L]
C(L) = 18(L + 100/L)

Use the First Derivative Test to find the value of L for which C(L) takes on its minimal value.

C'(L) = 18(1 - 100/L^2)
0 = 18(1 - 100/L^2)
0 = 1 - 100/L^2
100/L^2 = 1
100 = L^2
10 = L

The dimensions that minimize the cost are:

L = 10 m
w = 60/L = 60/10 = 6 m

The cost is:

C(10) = 18(10 + 100/10)
C(10) = 18(10 + 10)
C(10) = 18(20)
C(10) = 360

$360

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