CALCULUS PROBLEM HELP PLEASE

Question

A farmer wants to fence an area of 37.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be (in ft) in order to minimize the cost of the fence?

smaller value (ft)_____

larger value (ft) _____

Answer 1

____x_______x____

| | |

| | |

| y | y | y

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|________|________|

x x

Perimeter of the shape: P = 4x + 3y

Area of the shape: A = 2xy = 37.5

y = 18.75 / x solving for y in terms of x

P = 4x + 3(18.75 / x) replacing in perimeter expression

P = 4x + (56.25/x) = 4x + 56.25x^-1 perimeter in terms of x

P' = 4 - 56.25x^-2 derivative of perimeter

4 - 56.25x^-2 = 0 optimize

x² = 56.25/4

x = 3.75 million ft² (smaller value)

y = 18.75 / x = 18.75 / 3.75 = 5 million ft² (larger value)