What are the dimensions of the minimum cost to enclose 120 square feet?
A rectangular plot of land is to be fenced in using two kinds of fencing. Three sides will use fencing $4 a foot, while the remaining side use fencing $12 a foot.
x = length of the rectangle
y = width of the rectangle
C= total cost of the fence.
The equation for the cost is:
`C = 4(x+x+y) + 12y = 4(2x+y) + 12y = 8x + 4y + 12y`
`C= 8x + 16y`
Then, we need to express the right side as one variable. To do so, we are going to use the given area of the rectangle.
`120 = xy`
`120/x = y`
Substitute this to the cost equation.
`C = 8x + 12y = 8x + 16(120/x) `
`C= 8x + 1920/x`
Then, take the derivative of C.
`C' = 8x -1920/x^2`
`C' = (8x^2 - 1920)/x^2`
Set C' to zero. Solve for x.
`0 = (8x^2 - 1920)/x^2`
`0 = 8x^2-1920`
`0 = x^2 - 240`
`240 = x^2`
`+-sqrt240 = x`
`+-4sqrt15 = x`
Since x represent a dimension, take only the positive value.
`x = 4sqrt 15`
Substitute the value of x to y=120/x .
`y = 120/(4sqrt15) = 30/sqrt15 = (30sqrt15)/15 = 2sqrt15`
Hence the dimensions of the rectangular plot are:
Length = `4sqrt15 = 15.49 ft`
Width = `2sqrt15 = 7.75 ft.`