Question

A farmer needs to enclose an area of 800 square feet for an animal pen. one side of the fence is electric which is 3 times more expensive than regular fences. Find the dimensions of the cheapest design.

Answer 1

20ft by 40ft

Explanation:

The area of the fence is 800 ft². Let x represent one side of the fence and y the other side of the fence, hence:

xy = 800 (1)

The cost of fencing (C) is:

C = 3x + x + y + y

C = 4x + 2y (2)

From xy = 800, y = 800/x. Substituting in equation 2:

`C = 4x +2( (800)/(x))\\\\Minimum \ cost\ is\ at\ C'=0\\\\Hence:\\ \\C'=4-(1600)/(x^2) \\\\4-(1600)/(x^2) =0\\\\4=(1600)/(x^2) \\\\x=\sqrt{(1600)/(4) } \\\\x=20\ ft\\\\y=(800)/(x) =(800)/(20)=40\ ft`