Question 257304
a circular pool measure 14 feet across. one cubic yard of concrete is to be used to create a circular border of uniform width around the pool.
 if the border is to have a depth of 3 inches, how would will the border be
:
Change 1 cu yd to cu ft
3*3*3 = 27 cu/ft of concrete
Find the area covered if 3" thick: 3" = 3/12 = .25 ft
{{{27/.25}}} = 108 sq/ft
:
Let x = width of the concrete border
:
Pool radius = 7 ft (measures 14' across)
:
Overall area - pool area = 108 sq/ft
{{{pi*(x+7)^2}}} - {{{pi*7^2}}} = 108
:
{{{pi(x^2 + 14x + 49)}}} - (153.938) = 108
:
{{{pi(x^2 + 14x + 49)}}} = 108 + 153.938
:
{{{pi(x^2 + 14x + 49)}}} = 261.938
Divide both sides by Pi
x^2 + 14x + 49 = 83.377
:
x^2 + 14x + 49 - 83.377 = 0
:
x^2 + 14x - 34.377 = 0
Solve this using the quadratic formula
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
In this problem a=1; b=14; c=-34.377
{{{x = (-14 +- sqrt(14^2-4*1*-34.377 ))/(2*1) }}}
:
{{{x = (-14 +- sqrt(196-(-137.508) ))/2 }}}
:
{{{x = (-14 +- sqrt(333.508 ))/2 }}}
Positive solution is what we want here
{{{x = (-14 + 18.262)/2 }}}
x = {{{4.262/2}}}
x = 2.13 ft, the width of the concrete border (2' 1.6")
:
:
Check this solution on a calc, enter
(pi*9.13^2) - (pi*7^2) = 107.9 ~ 108 sq/ft
:
108 * .25 = 27 cu.ft which is 1 cu yd