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.
