A rectangular-shaped vegetable garden next to a barn is to be fenced on three sides with 120 total feet of fencing. Find the dimensions of the garden that will maximize the area.

The dimensions of the garden are

Length = 30 ft

Width = 60 ft

Explanation:

Given dа ta:

Total length of the fencing, L = 120 ft

Now,

let the length of the garden be 'x' feet

width of the garden be 'y' feet

Now, the area A for rectangle is given as:

A = xy

Now,

given y + 2x = 120 ft

or

y = 120 - 2x

substituting the value of y in the formula for area, we get

A = x * (120 - 2x)

or

A = 120x - 2x²

Now, for maximizing the area

dA/dx = 0

therefore, differentiating the formula for area with respect to side x

we get

dA/dx = 120 - 4x = 0

or

4x = 120

or

x = 30 ft

hence,

y = 120 - 2x

or

y = 120 - 2 (30)

or

y = 60 ft

Thus, the dimensions of the garden are

Length = 30 ft

Width = 60 ft