### Video Transcript

A hollow sphere of metal has an

internal radius of 1.8 centimeters and an external radius of 2.2 centimeters. One cubic centimeter of the metal

weighs 30 grams. Using the approximation 𝜋 equals

22 over seven, find the mass of the sphere.

In this example, we want to find

the mass of a hollow sphere, given the weight of the metal per cubic centimeter and

the internal and external radii of its metal shell. So let’s start by sketching the

sphere so that we can see what we’re looking for.

To find the volume of the solid

shell, which makes up the mass of the sphere, we subtract the volume of the inner

sphere, which has radius 1.8 centimeters, from the volume of the whole sphere,

that’s the sphere with radius 2.2 centimeters, the outer radius. And this will leave us with the

volume of the shell.

To do this, we recall that the

volume of a sphere of radius 𝑟 is given by four over three 𝜋𝑟 cubed. So now substituting in 2.2 for the

outer radius and 1.8 for the inner and taking out the common factor of four over

three 𝜋, we have the volume of the shell equal to four over three 𝜋 times 2.2

cubed minus 1.8 cubed. Evaluating our cubes and

subtracting leaves us with four over three 𝜋 times 4.816.

Now making some space for our next

calculation and making a note that the units of volume are cubic centimeters, we

have the volume of the outer shell equal to four over three 𝜋 times 4.816 cubic

centimeters. Now we want to find the mass of the

sphere. And we’re told that one cubic

centimeter of the metal shell weighs 30 grams. So for the mass of the sphere, we

multiply the volume we found by 30. And using the approximation 𝜋

equals 22 over seven, the mass of the sphere is 30 times four over three times 22

over seven times 4.816. This evaluates to 605.44, which we

round down to a whole number.

And so given the weight of the

metal as 30 grams per cubic centimeter, the internal radius of 1.8, and the outer

radius of 2.2 centimeters, we find that the mass of the hollow metal sphere is 605

grams.