Question

A rectangular pool is 12m long and 6m wide. A walkway of uniform width surrounds the pool. If the area of the walkway is 9m squared less than that of the pool, how wide is the walkway?

Answer 1

The walkway is 1.5 m wide.

The area of the pool is 12(6) = 72 m².

Adding a walkway of unknown width, x, around all 4 sides of the pool increases the width by 2x and the length by 2x; thus the area of the entire pool and walkway together would be given by

(12+2x)(6+2x)

We know that the area of just the walkway is 9 m² less than the area of the pool. This means that:

(12+2x)(6+2x)-72 = 72-9

Multiplying through we have:
12*6+12*2x+2x*6+2x*2x - 72 = 63
72 + 24x + 12x + 4x² - 72 = 63
24x + 12x + 4x² = 63
36x + 4x² = 63

Writing in standard form we have:
4x² + 36x = 63

We want to set it equal to 0 to solve, so subtract 63 from both sides:
4x² + 36x - 63 = 63 - 63
4x² + 36x - 63 = 0

Using the quadratic formula,


`x=(-b\pm √(b^2-4ac))/(2a) \\ \\=(-36\pm √(36^2-4(4)(-63)))/(2(4)) \\ \\=(-36\pm √(1296--1008))/(8)=(-36\pm √(1296+1008))/(8) \\ \\=(-36\pm √(2304))/(8)=(-36\pm 48)/(8)=(-36+48)/(8)\text{ or }(-36-48)/(8) \\ \\=(12)/(8) \text{ or }(-84)/(8)=1.5 \text{ or }-10.5`

Since a negative width makes no sense, the walkway is 1.5 m wide.