Problema Solution
A rectangular Playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 540ft of fencing is to be used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?
The smaller dimension is - x
The larger dimension is - x
The maximum area is - x sq feet
Answer provided by our tutors
The 540 feet of fencing is to be used around the four sides of the playground as well as another side to divide the playground. So, 540 feet = 2 * length + 3 * width.
The area of the playground is length * width.
The length of the playground can be written as 2 * length = 540 feet - 3 * width => length = 270 feet - 1.5 * width
The area is then width * (270 - 1.5 * width) = 270 * width - 1.5 * width^2
The width is maximized when the derivative of the area is 0, or 270 - 3 * width = 0
270 = 3 * width
width = 90
length = 270 - 1.5 * 90
The smaller dimension is - 90 feet
The larger dimension is - 135 feet
The maximum area is - 12,150 square feet