EXAMPLE 2.7.4

Gomer has been working out by lifting weights. He finds that a spherical lead-alloy weight with a radius of 3 inches weighs 20 pounds. He wishes to lift 100 pounds, so he special-orders a spherical weight with a radius of 15 inches. Why is Gomer in intensive care?

SOLUTION

Gomer is in intensive care because he tried to lift a weight that weighs much more than 100 pounds. To find the correct weight of the sphere that has the 15-inch radius, we can use the fact that the sphere with the 3-inch radius weighs 20 pounds to find the weight per cubic inch of one of these objects, and then apply that weigh factor to the 15-inch radius sphere. Recall the formula for the volume of a sphere:

Volume = (four thirds) times pi times (radius cubed)

The volume of the 3-inch radius sphere is

Volume = (four thirds) times pi times (3 cubed)

= (four thirds) times pi times (27)

= 113.1 cubic inches

Now find the volume of the sphere with the 15-inch radius:

Volume = (four thirds) times pi times (15 cubed)

= (four thirds) times pi times (3375)

= 14,137.2 cubic inches

Next, divide the volume of the large sphere by the volume of the small sphere:

(14,137.2 cubic inches) divided by (113.1 cubic inches) = 125

This means that the large spherical weight is 125 times as large as the small weight, so it weighs 125 times as much.

The large weight weighs 125 times (20 pounds) = 2,500 pounds.

Gomer thought that he was lifting a 100 pound weight, but in fact he was lifting a 2500 pound weight. The second weight wasn't 5 times heavier than the first weight. It was 125 times as heavy. This is because a sphere with a 15-inch radius is 125 times as large as a sphere with a 3-inch radius.