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Chapter 0: Problem 35

A pool measuring 10 meters by 20 meters is surrounded by a path of uniform width. If the area of the pool and the path combined is 600 square meters, what is the width of the path?

Short Answer

Expert verified
The width of the path is 5 meters.

Step by step solution

01

Calculate the area of the path

Subtract the area of the pool from the total area to get the area of the path alone. Thus, \[Area_{path} = Area_{total} - Area_{pool} = 600 - 200 = 400 \text{ square meters}.\]
02

Setup an equation

Since the path surrounds the pool and has a uniform width 'x', the length and the width of the combined area will be \(10 + 2x\) and \(20 + 2x\) respectively. The area of the pool and the path (combined), which is also equal to 600 square meters is given by \((10 + 2x )\times (20 + 2x) = 600.\]
03

Solve the equation for 'x'

Solve the quadratic equation \((10 + 2x )\times (20 + 2x) = 600\). This simplifies to \(4x^2 + 60x + 200 =600\). If you subtract 600 from both sides that gives you \(4x^2 + 60x -400 =0\). Dividing the equation by 4 yields \(x^2 + 15x -100 =0\). Factorizing this quadratic equation, we get \((x - 5)(x + 20) = 0\). This gives two roots, 'x = 5' and 'x = -20'. We can eliminate 'x = -20' as a path width cannot be negative. Thus, the width of the path is '5' meters.

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