Exercise 2 Page 652

Let's take a look at the given composite figure. In the graph we can see that it is a cylinder with a smaller cylinder hollowed out.

We are told that the volume of the composite figure is 3 300.7cm^3. Our goal is to check if this is true.

Notice that the volume of the figure V is a difference of two volumes — the volume of the outer cylinder V_1 and the the volume of the inner cylinder V_2. V = V_1-V_2 Let's now remember that the volume of a cylinder with the radius r and height h is calculated by using the following formula. V=π r^2 h Note that in the exercise we are given the diameters of the bases, not the radii. This will not be a problem, though. Let's now find V_1!

Volume of the Outer Cylinder

Since the diameter of the outer cylinder is 12 cm, its radius equals 12cm2 = 6 cm. Now that we know the radius and height of the cylinder, we can substitute them into the formula and calculate V_1.

The approximate volume of the outer cylinder is 2940.5 cubic centimeters.

Volume of the Inner Cylinder

Once again, we start by finding the radius of the base. Since the diameter of the inner cylinder is 4.2 cm, its radius equals 4.2cm2 = 2.1 cm. We can now substitute the values into the formula and calculate V_2.

The approximate volume of the inner cylinder is 360.2 cubic centimeters.

Volume of the Composite Figure

As we earlier said, the volume of the composite figure V is a difference of volumes V_1 and V_2. V = V_1-V_2 Let's substitute the found volumes into the expression and evaluate. V &= 2940.5- 360.2 &= 2580.3 ✓ Notice that the result is different from the answer from the problem. This is how we know that the problem is incorrect.

The most likely error is that the volumes were added, not subtracted. Notice that when we add volumes V_1 and V_2, we get the result from the problem. V &= 2940.5+ 360.2 &= 3300.7 *