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...

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.

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lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

Let, 

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.

`A =xy`

`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.`

 

 

 

 

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