A solid sphere of radius r is placed inside a hollow right circular cylinder of base radiu… — GRE General Test

A solid sphere of radius $r$ is placed inside a hollow right circular cylinder of base radius $r$ and height $3r$ such that the sphere rests on the bottom base of the cylinder. Water is then poured into the cylinder until the water level just covers the top of the sphere (a height of $2r$ ). What is the ratio of the volume of the empty space remaining in the cylinder above the water level to the volume of the water inside the cylinder?

A
$\frac{1}{3}$
B
$\frac{1}{2}$
C
$\frac{3}{4}$
$\frac{3}{2}$ Answer
E
$2$

Answer

The ratio of the volume of the empty space remaining in the cylinder above the water level to the volume of the water inside the cylinder is

\frac{3}{2}

The empty cylinder portion at the top has height

r

, giving a volume of

\pi r^3

. The submerged section up to height

2r

has total volume

2\pi r^3

, of which the sphere accounts for

\frac{4}{3}\pi r^3

. Subtracting the sphere's volume leaves

\frac{2}{3}\pi r^3

of water. The ratio of empty space to water volume is

\frac{\pi r^3}{\frac{2}{3}\pi r^3} = \frac{3}{2}

Step-by-Step Solution

Calculate the volume of the empty cylindrical space above the water height.

The total height of the cylinder is

3r

and the water level reaches height

2r

, leaving an empty height of

3r - 2r = r

. The volume of empty space is

V_{\text{empty}} = \pi r^2 \cdot r = \pi r^3

The empty space forms a smaller cylinder of base radius

r

and height

r

at the top.

Calculate the total volume of the cylindrical region filled up to the water height of

2r

V_{\text{region}} = \pi r^2 \cdot (2r) = 2\pi r^3

This represents the combined volume of the water and the submerged sphere.

Calculate the volume of the solid sphere of radius

r

V_{\text{sphere}} = \frac{4}{3}\pi r^3

Using the standard formula for the volume of a sphere,

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

Find the volume of the water by subtracting the sphere's volume from the total filled region.

V_{\text{water}} = 2\pi r^3 - \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3

The space taken up by the water is the total volume up to height

2r

minus the displacement volume of the sphere.

Compute the required ratio of

V_{\text{empty}}

V_{\text{water}}

\frac{V_{\text{empty}}}{V_{\text{water}}} = \frac{\pi r^3}{\frac{2}{3}\pi r^3} = \frac{1}{\frac{2}{3}} = \frac{3}{2}

Dividing the empty space volume by the water volume simplifies to

\frac{3}{2}

Key Concept

Displacement and volume calculations combining 3D geometric figures (cylinders and spheres).

Estimated Time:2m 30s

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