Exercise 14 Page 442

We are given a solid figure that has a cone and a hemisphere hollowed out of it. We want to find the volume of the remaining part of the solid.

Notice that the figure would be a cylinder if the hemisphere and cone were not hollowed out of it. This means that the volume of the figure V is the volume of the cylinder V_1 minus the volumes of the hemisphere V_2 and the cone V_3. V = V_1-V_2-V_3

Volume of the Cylinder

Let's start by finding the volume of the cylinder V_1.

To do so, we can use the formula for the volume of a cylinder. V_1=π r^2 h From the diagram we can see that the height h of the cylinder is 23 inches. Also, the radius of the base r is 6 inches. Let's substitute these values into the formula and calculate the volume.

The volume of the cylinder is about 2600 cubic inches.

Volume of the Cone

Let's now move on to finding the volume of a cone V_2.

For that, recall the formula for the volume of a cone. V_2= 13π r^2 h From the diagram we can see that the radius of the base r is 6 inches. We need to find the height of the cone h. Notice that the height of the cone and the radius of the hemisphere together add up to 23 inches.

Since we know that the radius of the hemisphere is 6 inches, we can find the height of the cone. h+6 = 23 ⇒ h = 17 The height of the cone is 17 inches. Let's substitute the values into the formula and calculate the volume.

The volume of the cone is about 641 cubic inches.

Volume of the Hemisphere

Finally, we have the hemisphere.

By definition, a hemisphere is half of a sphere. Now, the volume of a sphere is four thirds the product of π and the cube of the radius. V=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_3 of our hemisphere. V_3=1/2 * 4/3π r^3 From the diagram we can see that the radius is 6 inches. We can substitute this value for r in the above formula and simplify the right-hand side.

The volume of the hemisphere is about 452 cubic inches.

Volume of the Figure

Earlier we noted that the volume of the figure V is the volume of the cylinder V_1 decreased by the volumes of the cone V_2 and the hemisphere V_3. V = V_1- V_2- V_3 Let's substitute for the volumes with the values that we found. Then we will simplify the right-hand side. V = 2600- 641- 452 = 1507 The volume of the figure is about 1507 cubic inches.