.help with area

Question

Henry has 240

 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).

(a) Find a function that gives the area Ax

 of the garden (in square meters) in terms of x

A(x) =

(b) What side length x

 gives the maximum area that the garden can have?

side length x: ? meters

(c) What is the maximum area that the garden can have?

maximum area; ? square meters

.

Answer 1

Let x = length of the side parallel to the side without fencing and let y = length of each of the two sides perpendicular to the house.

x + 2y = 240

So, x = 240 - 2y

Area = A = xy = (240 - 2y)(y) = -2y² + 240y

Maximum area when A' = 0. -4y + 240 = 0 So, y = 60 m and x = 120 m.

Answer 2

240 meters = 2x +y

y = 240-2x

Area = A =xy = x(240-2x) = 240x-2x^2

A(x) = 240x-2x^2

A' = 240-4x=0

4x =240

x = 60 meters = length of each of 2 shorter sides

y = 240-2(60) =120 meters = length of the 3rd side

max A= 60(120) = 7,200 m^2