Question 161919
A circular pool measures 10 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 4 inches, how wide will the border be?
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AN interesting problem
Let x = width of the concrete border ( in feet)
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Find the the area of the concrete required rather than the volume
1 cu yd = 36" * 36" * 36" = 46646 cu/inches
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Divide this by the 4" thickness and we have the area
{{{46656/4}}} = 11664 sq/in is area of the concrete
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Change the area of the concrete to sq/ft
{{{11664/144}}} = 81 sq/ft is the area of the concrete
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Find the area of the pool: Ap = {{{pi*5^2}}} = 78.54 sq/ft
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Total area including the concrete border: 78.54 + 81 = 159.54 sq/ft
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Radius of the circular area including the border: (x+5)
{{{pi*(x+5)^2}}} = 159.54
x^2 + 10x + 25 = {{{159.54/pi}}}
x^2 + 10x + 25 = 50.78
x^2 + 10x + 25 - 50.78
A quadratic equation
x^2 + 10x - 25.78 = 0
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Solve this using the quadratic formula
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
In our equation: a=1; b=10; c=-25.78
{{{x = (-10 +- sqrt(10^2 - 4*1*-25.78 ))/(2*1) }}}
:
{{{x = (-10 +- sqrt(100 + 103.13 ))/(2) }}}
Do the math here and you will find the positive answer:
x = 2.12 ft is the width of the border
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Check our solution by finding the vol of concrete
Overall area - pool area = border area
{{{pi*(2.12+5)^2}}} - 78.54 =
159.26 - 78.54 = 80.72 sq/ft of concrete
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Change 80.72 to sq/in: 144 * 80.72 = 11623.68 sq/inches
Find the vol by multiplying by the thickness 4"
11623.68 * 4 = 46494.72 cu/in
Change to cu yds: {{{46495/46656}}} = .997 cu/yds ~ 1 cu/yd