Four hundred eighty dollars are available to fence in a rectangular garden. The fencing for the north and south sides of the garden costs $10 per foot and the fencing for the east and west sides costs $20 per foot. Find the dimensions of the largest possible garden.

Using x for the length of the east side (and is equal to the length of the west side) and y for the length of the north side (and is equal to the length of the south side), the equation that gives the total price equalized to 480 is:

[tex]20x+20x+10y+10y=480[/tex]

[tex]40x+20y=480[/tex]

Solving for y

[tex]y=\frac{-40x+480}{20}[/tex]

[tex]y=-2x+24[/tex]

The area of the garden is [tex]A=xy[/tex], to find the largest, substitute y in the formula of the area

[tex]A=x(-2x+24)=-2x^2+24x[/tex]

For the optimization, find the largest area, is needed the critical point. To find this point, derive A and equalize the derivative to zero:

[tex]A'=-4x+24=0[/tex]

Solve for x:

[tex]-4x=-24[/tex]

[tex]x=\frac{-24}{-4}[/tex]

[tex]x=6[/tex]

To see if x=6 is a maximum or a minimum, derive A' and substitute with x=6

[tex]A''=-4[/tex]

In this case, the second derivative of A doesn't depend on x, and it has a negative value, meaning the value found is a maximum. Using x=6 to find y

[tex]y=-2x+24[/tex]

[tex]y=-2(6)+24[/tex]

[tex]y=12[/tex]

The area is:

[tex]A=xy=6*12=72 ft^2[/tex]