Question

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.

Answer 1

The dimensions of the garden are

Length = 30 ft

Width = 60 ft

Step-by-step explanation:

Given data:

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