# Swimming Pool Border

• Jan 13th 2007, 08:53 AM
symmetry
Swimming Pool Border
A pool in the shape of a circle 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?

NOTE: 1 cubic yard = 27 cubic feet.

I believe we use a volume formula here, right?

Aslo, what is meant by the math phrase "uniform width"? I always see uniform width in geometry books.
• Jan 13th 2007, 09:43 AM
CaptainBlack
Quote:

Originally Posted by symmetry
A pool in the shape of a circle 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?

NOTE: 1 cubic yard = 27 cubic feet.

I believe we use a voluem formula here, right?

Aslo, what is meant by the math phrase "uniform width"? I always see uniform width in geometry books.

Uniform width means that it is the same width all around the pool, so the
inner and outer edges constitute two concentric circles.

If the width of the border is $\displaystyle b\ \mbox{ft}$, the the area of the border is:

$\displaystyle A=\pi (b+5)^2 - \pi\, 5^2$

That is it is the difference of area between the circle of radius $\displaystyle b+5$ and one of radius $\displaystyle 5 \mbox{ ft}$.

Then the volume of cement required is:

$\displaystyle V=A/3 \mbox{ cubic ft}$

as $\displaystyle 4 \mbox{ inches}=1/3 \mbox{ ft}$,

so. as $\displaystyle V=1 \mbox{ cubic yd}=27 \mbox{ cubic ft}$ we have:

$\displaystyle (\pi (b+5)^2 - \pi\, 5^2)/3=27$

rearranging this gives:

$\displaystyle b^2+\frac{10}{\pi}b-\frac{27}{\pi}=0$.

This is a quadratic in $\displaystyle b$ which can be solved using the quadratic formula to give:

$\displaystyle b=\frac{\sqrt{27 \pi+25}-5}{\pi}$

or:

$\displaystyle b=-\frac{\sqrt{27 \pi+25}+5}{\pi}$

The second of these is negative and so cannot be the solution which leaves the first, so:

$\displaystyle b=\frac{\sqrt{27 \pi+25}-5}{\pi} \approx 1.744 \mbox{ ft}$

RonL
• Jan 13th 2007, 01:45 PM
symmetry
ok
I really enjoy following your steps when reading through the questions and replies.