Question

Which best describes the relationship among the volumes of hemisphere X, cylinder Y, and cone Z?

A. The sum of the volumes of X and Y equals the volume of Z.

B. The sum of the volume of X and Z equals the volume of Y.

C. The difference of the volumes of Z and Y equals the volume of X.

D. The difference of the volumes of X and Z equals the volume of Y.

Answer 1

Answer:

B. The sum of the volume of X and Z equals the volume of Y.

Step-by-step explanation:

To solve the problem, we calculate the volume of the three solids, and then we compare them.

The volume of X is the volume of a hemisphere of radius r, which is half the volume of a sphere of radius r, so:

$V_x = \frac{1}{2}(\frac{4}{3}\pi r^3)=\frac{2}{3}\pi r^3$

The volume of Y is the volume of a cylinder of radius r and height

h = r

so it is given by the formula:

$V_Y=\pi r^2 h = \pi r^2 \cdot r = \pi r^3$

Finally, the volume of Z is the volume of a cone of radius r and height

h = r

So the volume is given by

$V_Z = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi r^2 \cdot r = \frac{1}{3}\pi r^3$

So we see that the correct option is

B. The sum of the volume of X and Z equals the volume of Y.

In fact:

$V_X + V_Z = \frac{2}{3}\pi r^3 + \frac{1}{3}\pi r^3 = \pi r^3 = V_Y$