Question

A marble top bathroom sink has the same shape of a hemispherical (a half-sphere) porcelain bowl with an inside diameter of 11 inches. how much water, in gallons, will the bowl hold? Round to the nearest gallon as needed.

Answer 1

So what we need to do here is find the volume of the bowl in cubic inches and then convert it to gallons. For this purpose is important to recal that:

$1gal=231in^3$

So let's find the volume of the bowl in cubic inches first. The volume of a sphere with a radius r is given by:

$V=\frac{4}{3}\pi\cdot r^3$

In this case, the bowl is a half-sphere so it has half the volume:

$V=\frac{\frac{4}{3}\pi\cdot r^3}{2}=\frac{2}{3}\pi\cdot r^3$

The radius is half the diameter so:

$r=\frac{11in}{2}=5.5in$

Then the volume of the bowl in cubic inches is:

$V=\frac{2}{3}\pi\cdot r^3=\frac{2}{3}\pi\cdot(5.5in)^3=348.45in^3$

Now using the convertion factor between gallons and cubic inches and the rule of three we get:

$\begin{gathered} 231in^3------1gal \\ 348.45in^3----x \\ x=\frac{348.45in^3\cdot1gal}{231in^3}=1.5\text{gal} \end{gathered}$

And we need to round our answer to the nearest gallon which means that the answer is 2 gallons.