a circluar pool is 10 ft across.one cubic yard of concrete is used to create a border of uniform width around. the border is to have depth of 4 in, how wide?

Question

this question is a word problem from my algebra book in the section quadratic equations

Answer 1

Let's do everything in inch units:

radius of pool = 5 feet = 60 inches.

Volume of border = 1 cubic yard = 36³ cubic inches.

Area of border = Volume/thickness = 36³/4 = 11664 square inches.

 

Let the width of the border be B inches, so that the border goes from radius = 60 inches to radius = 60+B inches.

The Area of the border is:

  A_B = pi (60+B)² - pi (60)²

       = pi(60² + 120B + B² - 60²)

so

  A_B/pi = 120B + B²

or in standard quadratic form and replacing Ab/pi with 11664/pi ~ 3713

  B² + 120B - 3713 = 0

Using the quadratic formula:

  B = [ -120 +/- sqrt( 120² - 4*1*(-3713) ) ] / (2*1)

     = -60 +/- sqrt(120²/4 + 3713)

     = -60 +/- sqrt(3600 + 3713)

     = -60 +/- 85.52

We take the positive solution:

 

  B = 25.52 inches. <-- width of border

 

Checking:

 Area of border = pi(60+25.52)² - pi(60)² ~ 11667 square inches

 Volume of border = 4 inches * 11667 sq in = 46667 cu in, convert to cubic yards:

       46667/(36³) = 1.0002    OK !