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, 

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

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