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A cylindrical cup is used to fill a bowl, which is in the sh

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A cylindrical cup is used to fill a bowl, which is in the sh [#permalink] New post 26 Oct 2017, 02:23
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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

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[Reveal] Spoiler: OA
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Re: A cylindrical cup is used to fill a bowl, which is in the sh [#permalink] New post 26 Oct 2017, 04:53
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Let's use letter c for cup and letter e for hemisphere. The information above can be rewritten as \(r_c = \frac{1}{2}r_e\) and \(h_c = 2r_c\).

The volume of the cup, given that it is a cylinder is equal to \(\pi r_c^2 h_c\), while the volume of the hemisphere is half the volume of a sphere, i.e. \(\frac{1}{2}(\frac{4}{3}\pi r_e^3) = \frac{2}{3}\pi r_e^3\). Substituting in the formula for the volume of the cup, using the information given, leads to \(\pi \frac{1}{4}r_e^2 r_e = \frac{1}{4} \pi r_e^3\).

Now we can multiply the volume of the cup by our choices to get the one which does not exceed the volume of the hemisphere.

If we use 2 cups, the volume becomes \(\frac{1}{2} \pi r_e^3\), which is less than \(= \frac{2}{3}\pi r_e^3\).
If we use 3 cups, the volume becomes \(\frac{3}{4} \pi r_e^3\), which is higher than \(= \frac{2}{3}\pi r_e^3\).

Answer A
Re: A cylindrical cup is used to fill a bowl, which is in the sh   [#permalink] 26 Oct 2017, 04:53
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A cylindrical cup is used to fill a bowl, which is in the sh

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