Video Transcript
A new roll of toilet paper has 500 sheets and a diameter of 11 centimeters. An empty roll has a diameter of 4.5 centimeters. Estimate how many sheets are left on a roll with diameter seven centimeters.
We are told that a toilet roll of diameter 11 centimeters has 500 sheets. The diameter of the empty roll is 4.5 centimeters. We can calculate the volume of toilet paper that corresponds to 500 sheets by
subtracting the volume of the empty roll from the volume of the full roll.
We recall that the volume of a cylinder is equal to 𝜋𝑟 squared ℎ, where 𝑟 is the
radius and ℎ is the height of the cylinder. The volume of the full roll is therefore equal to 𝜋 multiplied by 5.5 squared
multiplied by ℎ, since the radius is half of the diameter and one-half of 11
centimeters is 5.5 centimeters. In the same way, the volume of the empty roll is 𝜋 multiplied by 2.25 squared
multiplied by ℎ, noting that the height ℎ is the same unknown value in both
cases.
Since 5.5 squared is equal to 30.25 and 2.25 squared is equal to 5.0625, the volume
of toilet paper is equal to 30.25𝜋ℎ minus 5.0625𝜋ℎ. This is equal to 25.1875𝜋ℎ cubic centimeters and is the volume of paper that
corresponds to 500 sheets. In order to answer this question, we need to calculate the number of sheets on a
toilet roll with a diameter of seven centimeters. The volume of this new roll will be equal to 𝜋 multiplied by 3.5 squared multiplied
by ℎ, since the radius is equal to 3.5 centimeters.
Since the volume of the empty roll will be the same, we can calculate the volume of
toilet paper remaining by subtracting 5.0625𝜋ℎ from 12.25𝜋ℎ, as 3.5 squared is
equal to 12.25. This simplifies to 7.1875𝜋ℎ.
After clearing some space, we can now calculate how many sheets 𝑥 this corresponds
to. We know that the ratio of the two volumes will be approximately equal to the ratio of
the number of sheets. Setting them equal to one another to solve the equation, this means that 7.1875𝜋ℎ
over 25.1875𝜋ℎ is equal to 𝑥 over 500. On the left-hand side, we can cancel the shared factors of 𝜋 and ℎ from the
numerator and denominator. 7.1875 over 25.1875 is equal to 115 over 403. We can then multiply through by 500 so that 𝑥 is equal 115 over 403 multiplied by
500, which is equal to 142.6799 and so on.
We can therefore estimate that there are 143 sheets left on the toilet roll when its
diameter is seven centimeters.
