Exercise 12 Page 442

We want to find the volume of the given composite solid.

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

1. Find the volume of the hemisphere.
2. Find the volume of the cylinder.
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, this is the formula for the volume V_h of a hemisphere. V_h=1/2 * 4/3π r^3 From the diagram we can see the radius equals 4 meters. We can substitute this value for r in the formula. Then we will simplify the right-hand side.

The volume of the hemisphere is about 134 cubic meters.

Volume of the Cylinder

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

To do so, we can use the formula for the volume of a cylinder. V_c=π r^2 h From the diagram we can see that the height h of the cylinder is 17 meters and that the radius of the base r is 4 meters. Let's substitute these values into the formula and calculate the volume.

The volume of the cylinder is 854 cubic meters.

Volume of the Figure

Earlier we noted that the volume of the figure is the sum of the volumes of the hemisphere and the cylinder. V = V_h+ V_c Let's substitute the volumes for the values that we found and add them. V = 134+ 854 = 988 The volume of the figure is about 988 cubic meters.