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.
A cylindrical cup is used to fill a bowl
This topic has expert replies
- EconomistGMATTutor
- GMAT Instructor
- Posts: 555
- Joined: Wed Oct 04, 2017 4:18 pm
- Thanked: 180 times
- Followed by:12 members
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.
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.
GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.