Question 1207439: 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 thepool.If the border is to have a depth of 3 inches,how wide will the border be? (1 cubic yard 27 cubic feet).
I need the set up here.
Thanks.
Found 4 solutions by math_tutor2020, MathTherapy, Theo, ikleyn:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
w = width of the walkway border around the pool
w > 0 since a negative width doesn't make sense. Neither does a width of 0.
The pool has a 10 ft diameter.
Cut that in half to get a 5 ft radius.
The area of the pool itself, ignoring the outer ring walkway, is
A = pi*r^2
A = pi*5^2
A = 25pi
Now accounting for the walkway, the 5 ft radius bumps up to 5+w feet.
Let's find the area of the larger circle that includes the surrounding walkway.
A = pi*(radius)^2
A = pi*(5+w)^2
A = pi*(25+10w+w^2)
Subtract the two circle areas to get the area of the outer ring only.
ringArea = largerCircleArea - smallerCircleArea
ringArea = pi*(25+10w+w^2) - 25pi
ringArea = pi*(25+10w+w^2-25)
ringArea = pi*(w^2+10w)
Multiply this ring area by the depth, or height, of the walkway.
This will get us the volume of concrete needed to make the walkway.
height = 3 inches = 3/12 of a foot
area*height = volume
pi*(w^2+10w)*(3/12) = 27 is one possible setup equation.
I'll let the student solve this equation for variable w.
You will need the quadratic formula at some point.
The quadratic formula will yield one positive solution and one negative solution.
Ignore the negative solution since we stated that w > 0 at the top.
Hint: The approximate final answer is somewhere between 2.5 and 3.1 feet.
Answer by MathTherapy(10552)  (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! i think the way to answer this is to find the volume of the outer cylinder formed and subtract the volume of the inner cylinder formed.
the result will be the volume of the border.
the volume of a cylinder is equal to pi * r^2 * h
r is the radius of the base of the cylinder.
h is the height of the cylinder.
the height of both cylinders will be 4 inches which is equal to 1/3 feet.
the volume of the inner cylinder is equal to pi * 5^2 * 1/3.
since the diameter of the inner cylinder is the same as the diameter of the pool, which is 10 feet, the radius of the inner cylinder is 5 feet.
the radius of the outer cylinder will be 5 plus the width of the border.
if we let x = the width of the border, then the radius of the outer cylinder is equal to (5 + x).
the volume of the outer cylinder is equal to pi * r * h which is equal to pi * (5 + x)^2 * 1/3.
the volume of the border is equal to the volume of the outer cylinder minus the volume of the inner cylinder.
that becomes 1/3 * pi * (5 + x)^2 minus 1/3 * pi * 5^2.
factor out the 1/3 * pi to get:
1/3 * pi * ((5 + x)^2 - 5^2)
simplify to get 1/3 * pi * (25 + 10x + x^2 - 25)
simplify further and order the terms in descending order of degree to get:
1/3 * pi * (x^2 + 10x)
that's the volume of the border.
x is the width of the border.
since the volume of the border is equal to 27 cubic feet, you get:
1/3 * pi * (x^2 + 10x) = 27
divide both sides of this equation by 1/3 * pi to get:
x^2 + 10x = 27 / (1/3 * pi) which becomes:
x^2 + 10x = 25.78310078.
subtract 25.78310078 from this equation to get:
x^2 + 10x - 25.78310078 = 0
factor to get x = 2.1262262650017.
that's the width of the border and should be your answer if i did this right.
that's one of the roots.
the other root is negative, so it is discarded as a possibility.
i used a quadratic equation solver to get that answer.
you only asked for the setup.
i solved it to make sure the setup would yield an answer that made sense.
try to solve it yourself to see if you get the same answer, or at least something close to it.
the setup i chose is to create cylinders with a height of 4 inches and then find the volume of the outer cylinder and the volume of the inner cylinder and then subtract the volume of the inner cylinder from the volume of the outer cylinder to get the volume of the border.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
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 3 inches, how wide will the border be? (1 cubic yard is 27 cubic feet).
I need the set up here.
Thanks.
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In the post by @Theo, everything is WRONG.
It is wrong, since @Theo systematically uses in his post incorrect value of 1/3 of a foot
as the height of the cylinders, which corresponds to 4 inches.
The correct value of the height of the cylinders in this problem is 3 inches (given),
which is 1/4 of a foot.
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Since radius of smaller circle (pool) is 5' (