Question

I would like to create a rectangular orchid garden that abuts my house so that the house itself forms the northern boundary. The fencing for the southern boundary costs $4 per foot, and the fencing for the east and west sides costs $2 per foot. If I have a budget of $120 for the project, what are the dimensions of the garden with the largest area I can enclose

Answer 1

x = 15 ft

y = 15 ft

A(max) = 225 ft²

Explanation:

Let call "x " and " y " sides of the rectangle x (side paralll to the northern boundary, then:

A(r) = x*y and 4*x + 2*2*y = 120 or 4*x + 4*y = 120

4*x + 4*y = 120 ⇒ x + y = 30 ⇒ y = 30 - x

Area of the garden as a function of x is:

A(x) = x* ( 30 - x ) ⇒ A(x) = 30*x - x²

Taking derivatives on both sides of the equation

A´(x) = 30 - 2*x

A´(x) = 0 ⇒ 30 - 2*x = 0

2*x = 30

x = 30/2

x = 15 ft

And y = ( 30 - x )

y = 15 ft

A(max) = 15*15

A(max) = 225 ft²