Exercise 9 Page 441

We want to find the volume of the following composite figure.

We see that the solid is the combination of two distinct solids, a hemisphere and a cone. Here is what we need to do to find its volume.

1. Find the volume of the hemisphere.
2. Find the volume of the cone.
3. Add the volumes.

Let's do it!

Volume of the Hemisphere

Let's start with the hemisphere.

A hemisphere is half of a sphere. Now, the volume of a sphere V_s is four thirds the product of π and the cube of the radius. V_s=4/3π r^3 Since the volume of a hemisphere is half the volume of a sphere, the formula for the volume of a sphere can be adjusted slightly to find the volume V_h of a hemisphere. V_h=1/2 * 4/3π r^3 From the diagram we can see that the diameter of the hemisphere is 14 centimeters. This means that the radius is 14÷ 2=7 centimeters. We can substitute this value for r in the formula and simplify the right-hand side.

The volume of the hemisphere is about 718 cubic centimeters.

Volume of the Cone

Next, let's consider the cone. We want to find its volume V_c.

To do so, we can use the formula for the volume of a cone. V_c= 13π r^2 h From the graph we can see that the height h of the cone is 17 centimeters, and we already found that the radius of the base r is 7 centimeters. Let's substitute these values into the formula and calculate the volume.

The volume of the cone is about 872 cubic centimeters.

Volume of the Figure

Earlier we noted that the volume of the figure is the sum of the volumes of the hemisphere and the cone. V = V_h+ V_c Let's substitute for the volumes with the values that we found and add them. V = 718+ 872 = 1590 The volume of the figure is about 1590 cubic centimeters.