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A cylindrical cup is used to fill a bowl

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by VJesus12 » Fri Dec 08, 2017 6:57 am
A cylindrical cup is used to fill a bowl, which is in the shape of a hemisphere (half of a sphere), with milk. If the radius of the cup is half the radius of the hemisphere, and the height of the cup is double its radius, what is the maximum number of cups that can fill the bowl without overflowing it?

A. 2 cups
B. 3 cups
C. 5 cups
D. 6 cups
E. 9 cups

The OA is A.

This is a hard question to me. How can I solve it? Can any expert give a good explanation of this? Thanks in advanced.
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by EconomistGMATTutor » Fri Dec 08, 2017 12:10 pm
Hello Vjesus12.

Let's take a look at your question.

Volume of cylindrical cup: $$V_{cy}\ =\ \pi\cdot r^2\cdot h.$$ Volume of a hemisphere (bowl): $$V_{he}=\frac{2}{3}\cdot\pi\cdot R^3.$$ We have $$r=\frac{R}{2}\ \ \ \ and\ \ \ \ h=2\cdot r\ \leftrightarrow\ \ h=R.$$ Number of cups $$\frac{V_{he}}{V_{cy}}=\frac{\frac{2}{3}\pi\cdot R^3}{\pi\cdot r^2\cdot h}=\frac{\frac{2}{3}\pi\cdot R^3}{\pi\cdot\frac{R^2}{4}\cdot R}=\frac{\frac{2}{3}R^3}{\frac{R^3}{4}}=\frac{8}{3}\approx2.66667.$$ So, the maximum number of cups that can fill the bowl without overflowing is 2.

When the third cup is poured, the milk overflows.

So, the correct answer is A.

I hope this explanation may help you.

I'm available if you'd like a follow up.

Regards.
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