Question

Tom has 120 meters of fencing. He will use it to form three sides of a rectangular garden. The fourth side will be along a house and will not need fencing. As shown below, one of the sides has length x (in meters).
Find a function:
What is the side length?
What is the maximum area?

Answer 1

Answer:

a) Function A(x) =  (60*x - 1/2*x²)

b)Side length     x = 60 m      the other side   y = 30 m

c) A (max)  = 1800 m²

Step-by-step explanation:

Area of rectangular garden with length  " x " and wide "y"

A = x*y        

Perimeter of rectangular area ( only three sides 2*y and 1*x)

P(r)  = 2*y + x

P(r) = 120 = 2*y + x

y = ( 120 - x ) /2

Area of the garden as function of x is

A(x) = [( 120 - x ) /2]*x          ⇒     A(x) = (60*x - 1/2*x²)

Taking derivatives on both sides of the equation:

A´(x) = 60 - x

A´(x)  = 0            60  -  x  = 0

x = 60 m

Then

y = ( 120  -  x ) / 2                ⇒   y = (120 - 60 )/2

y = 30 m

A(max) = 30 * 60

A(max) = 1800 m²