Question

A gardener has 120 ft of fencing to fence in a rectangular garden. one side of the garden is bordered by a river and so it does not need any fencing.

1. what dimensions would guarantee a garden with an area of 1350 ft^2?
2. What dimensions would guarantee the greatest area? how much is the greatest area?

Answer 1

two width, each of x
one length of y
y + 2x = 120
y = 120 - 2x

area = xy
= x(120-2x)
= 120x - 2x^2

1. 120x - 2x^2 = 1350
x^2 - 60x + 675 = 0
(x - 45)(x + 15) = 0
x = 45 or x = -15, the last we will reject

if x = 45 , y = 30

2. area = 120 - 2x^2 is a parabola which opens downwards so the vertex will be the maximum .
the x of the vertex is -120/(-4) = 30
when x = 30
y = 120-60 = 60

A width of 30 and a length of 60 ft will produce the largest area of 1800 ft^2