A rancher wants to enclose a rectangular region with fencing and also divide the

Question

A rancher wants to enclose a rectangular region with fencing and also divide the region in half with a length of fence. In other words, the region will look like this: We want o find the area of the largest region that can be enclosed in this way if the rancher only has 1200 ft of fencing. In the text box below answer the following questions. a. Let X be the length of the three vertical lengths of fencing and Y the length of the two horizontal pieces of fencing. Write down an equation for the objective function in terms of X and Y. b. Write down the constraint equation in terms of X and Y. c. Write your objective function in terms of just the variable X. d. What value of X results in the maximum area?

Explanation / Answer

A.

Objective function will be area of field

A = X*Y

B.

constraint equation will be total available wire for fencing,

P = Length of total field

P = 3X + 2Y = 1200 ft

3X + 2Y = 1200

C.

3X + 2Y = 1200

Y = (1200 - 3X)/2

So, area will be

A = X*(1200 - 3x)/2

A = 600X - 3X^2/2

D.

for maximum area, dA/dX = 0

dA/dX = 600 - 3*2*x/2 = 0

600 - 3X = 0

X = 600/3 = 200 ft

Y = (1200 - 3X)/2 = (1200 - 3*200)/2

Y = 300 ft

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