Question

A plan for a park has a rectangular plot of wild flowers that needs to be enclosed by 54 feet of fencing. Only three sides need to be enclosed because one side is bordered by the parking lot.

1. What is the largest area possible for the garden?

2. What width will produce the maximum area?

3. What is the length of the garden that will produce the maximum area?

Answer 1

The largest area possible for the garden is 364.5 ft²

The width that will produce the maximum area is 13.5 ft

The length that will produce the maximum area is 27 ft

Area of a rectangle:

• area = lw

where

l = length

w = width

Only three sides need to be enclosed because one side is bordered by the parking lot. Therefore,

perimeter = l + 2w

54  = l + 2w

l = 54 - 2w

Therefore,

area = (54 - 2w)w

(54 - 2w)w = 0

w = 0 or 27

average = 27 /2 = 13.5

Hence, the max area is at w = 13.5 meters

Therefore,

l = 54 - 2(13.5) = 27 ft

length = 27 ft

width =  13.5 ft

Therefore,

area = 27 × 13.5 = 364.5 ft²

Therefore, the largest area possible for the garden is 364.5 ft²

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