A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. 648 feet of fencing is used. Find the dimensions of the playground?

Answer 1

Without further information it is impossible to say what the dimensions are; only an equation can be given relating the width to length which will have infinitely many solutions as there are 2 unknowns:

Let one dimension of the playground be x and the other be y; let the extra fence be constructed parallel to the side of length x. Then:

total fence used = perimeter of rectangle + fence across the middle

→ 648 ft = 2x + 2y + x

→ 2y + 3x = 648 ft

As we only have 1 equation but 2 unknowns, there are infinitely many possible solutions, eg:

2 ft wide by 321 ft long, split with a fence parallel to the 2 ft side;

4 ft wide by 318 ft long, split with a fence parallel to the 4 ft side;

129 ft wide by 130 ft long, split with a fence parallel to the 130 ft side.

If the problem included other information, for example: the area of the playground is a maximum, then:

From above:

2y + 3x = 648 → y = 324 - 3x/2

and area = xy

→ area = x(324 - 3x/2)

→ area = 324x - 3x²/2

To find the maximum area, completing the square gives:

area = 3/2 (216x - x²)

→ area = -3/2 (x² - 216x)

→ area = -3/2 ((x - 108)² - 108²)

→ area = 3/2 (108² - (x - 108)²)

As x varies (x - 108)² is greater than or equal to 0, and the greater it is the smaller the area is; the maximum area is when (x - 108)² is a minimum, ie:

(x - 108)² = 0

→ x - 108 = 0

→ x = 108

→ y = 324 - 3 × 108 / 2 = 162

Thus the maximum area is when the playground is 162 ft long by 108 ft wide with the dividing fence parallel to the 108 ft side.