A right circular cylindrical container with a base radius of R inches is partially filled … — SAT

A right circular cylindrical container with a base radius of $R$ inches is partially filled with water. A solid metal sphere with a radius of $r$ inches is placed into the container and becomes completely submerged, causing the water level to rise by $1.5$ inches without any water overflowing. The sphere is then removed, and a solid right circular cone with a base radius of $r$ inches and a height of $12$ inches is placed vertex-down into the container. When the cone is completely submerged, the water level is $1$ inch higher than the container's original water level. What is the value of $R$ , in inches?

Answer: 9 inches

Answer

The radius of the cylinder,

R

, is

9

inches.

The volume of a submerged solid is equal to the volume of the cylinder of water it displaces. By setting up equations for the sphere and the cone, we get

\frac{4}{3}r^3 = 1.5 R^2

and

4r^2 = R^2

. Substituting the second equation into the first yields

r = 4.5

, which then gives

R = 2r = 9

Step-by-Step Solution

Equate the volume of the sphere to the volume of water it displaces in the cylinder.

\frac{4}{3}\pi r^3 = 1.5\pi R^2 \implies \frac{4}{3}r^3 = 1.5 R^2

The volume of a submerged solid equals the volume of the fluid it displaces. The displaced fluid takes the shape of a cylinder of radius

R

and height equal to the water level rise (

1.5

inches).

Equate the volume of the cone to the volume of water it displaces in the cylinder.

\frac{1}{3}\pi r^2 (12) = 1.0\pi R^2 \implies 4r^2 = R^2

Similarly, the volume of the cone is equal to the volume of a cylinder of radius

R

and height equal to the water level rise (

1

inch).

Solve the system of equations by substituting

R^2

into the sphere's displacement equation.

\frac{4}{3}r^3 = 1.5(4r^2) \implies \frac{4}{3}r^3 = 6r^2 \implies r = 4.5

By replacing

R^2

with

4r^2

, we reduce the system of equations to a single equation containing only

r

. Since

r \neq 0

, we can divide by

r^2

to solve for

r

directly.

Calculate the value of

R

from the relationship between

R

and

r

R = 2r = 2(4.5) = 9

Since

R^2 = 4r^2

and radii must be positive quantities,

R = 2r

Key Concept

Using water displacement to relate the volumes of three-dimensional geometric solids (cylinders, spheres, cones) and solving non-linear systems of equations.

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