Question

You need to enclose a rectangular plot of land where three sides are fence and one side is stone wall. the cost of the fence is $4 per foot, the cost of the stone wall is $16 per foot. the area of the land enclosed must be 40 ft^2 what are the dimensions (x and y so that the cost of enclosing the land is minimized?

Answer 1

Let x be the lengths parallel to the brick side and y be the lengths perpendicular to the brick side...

C=4(2y)+4x+16x

C=8y+20x

The area of this enclosure is xy and we are told that it is equal to 40 ft^2 so:

xy=40, so we can say y=40/x, using this y in our Cost function gives us:

C=8(40/x)+20x

C=320/x+20x

C=(320+20x^2)/x  now we can take the derivatives...

dC/dx=(40x^2-20x^2-320)/x^2

dC/dx=(20x^2-320)/x^2

d2C/dx2=(40x^3-40x^3+640x)/x^4

d2C/dx2=640/x^3

Since x>0, the acceleration will always be positive, which means than when dC/dx=0, it will represent an absolute minimum for C(x)...

dC/dx=0 when 20x^2-320=0, x^2=16, x=4

So the sides parallel to the brick wall are 4 feet and those perpendicular to the brick wall are 10 feet when the cost is minimized