GRE® Solid Geometry Practice Question

Concept: Volume of cylinder; volume of sphere

This GRE quant practice question is a solid geometry problem solving question. A classic example of equating volume of water in a vessel getting displaced to the volume of the object immersed in the vessel. Volumes of cylinder and sphere.

Question: A cylindrical vessel is filled with water up to some height. If a sphere of diameter 8 cm is dropped into the cylinder, the water level rises by half of the initial level. Instead, if a sphere of diameter 16 cm is dropped, the water level rises to a height h2. What percentage of this new height h2 is the initial level of water?

  1. 33⅓%
  2. 40%
  3. 20%
  4. 25%
  5. 16⅙%

Explanatory Answer

Video Explanation will be added soon

3 steps to the answer

Use these hints to get the answer

  1. The volume of water displaced when sphere 1 is dropped will be equal to the volume of sphere 1. Frame an equation equating the volumes and determine the displaced height.
  2. The volume of water displaced when sphere 2 is dropped will be equal to the volume of sphere 2. Frame an equation equating the volumes.
  3. Using the two equations, of volume displaced compute the relation between initial height of water and the new height of water h2.

Step 1: Determine the height to which water is displaced when sphere 1 is dropped

Let the radius of the cylinder be 'r'.
Let the initial height up to which water is filled be 'h'.
Let the height upto which water is filled after the sphere of diameter 8 cm is dropped be h1.
According to the question, the water level rises by 50% after sphere 1 is dropped into the cylinder.
Therefore, h1 - h = 0.5h

The water is stored in a cylindrical vessel. So, it will take the shape of the vessel that holds it.
The volume of water displaced = π r2 (height of water displaced)
= π r2 (0.5h)

The volume of water displaced is the same as the volume of sphere 1.
Volume of sphere = \\frac{4}{3})πr3
The diameter of the sphere = 8 cm. So, its radius is 4 cm.
The volume of the sphere = \\frac{4}{3})π(4)3
Equating the two volumes we get π r2 (0.5h) = \\frac{4}{3})π(4)3 .... equation 1

Step 2: Determine the height to which water is displaced when sphere 2 is dropped

The height of water in the vessel after sphere of diameter 16 cm is dropped = h2
Height of water displaced after sphere 2 is dropped = h2 - h.
Therefore, volume of water displaced after sphere 2 is dropped = π r2 (h2 - h)

The volume of water displaced is the same as the volume of sphere 2.
The diameter of the sphere = 16 cm. So, its radius is 8 cm.
The volume of the sphere = \\frac{4}{3}) x π x 83
Equating the two volumes we get π r2 (h2 - h) = \\frac{4}{3}) x π x 83 .... equation 2

Step 3: Solve the 2 equations obtained in step 1 and step 2

Divide equation (2) by equation (1)
\\frac {{\pi }{{r}^{2}\left ( {{{h}_{2}-h}} \right )}} {{\pi }{{r}^{2}\left ( {{0.5h}} \right )}}) = \\frac {\frac {4} {3}{\pi }{{8}^{3}}} {\frac {4} {3}{\pi }{{4}^{3}}})
\\frac{h_2 - h}{0.5h}) = 8
(h2 - h) = 4h
h2 = 5h
Or h = \\frac{1}{5}) h2
i.e., h = 20% of h2

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