Loo roll emergency
When the roll of toilet paper is half as wide, what percentage of the paper is left?
Age
14 to 16
Challenge level
Problem
New toilet rolls are cylindrical, 12 cm in diameter, and the cardboard tube in the middle is 4 cm in diameter.
Image
My roll has diameter 6 cm, so it looks about half the size.
What percentage of the toilet paper is left?
Student Solutions
The amount of toilet paper left will be proportional to the area of the paper on the cross-section of the roll. The cross sections of the two rolls are shown below.
On the new roll, this area is $\pi\times6^2-\pi\times2^2=36\pi-4\pi=32\pi$ cm$^2.$
On my roll, this area is $\pi\times3^2-\pi\times2^2=9\pi-4\pi = 5\pi$ cm$^2.$
So $\frac{5}{32}=15.625\%$ of the toilet paper is left.
Image
On the new roll, this area is $\pi\times6^2-\pi\times2^2=36\pi-4\pi=32\pi$ cm$^2.$
On my roll, this area is $\pi\times3^2-\pi\times2^2=9\pi-4\pi = 5\pi$ cm$^2.$
So $\frac{5}{32}=15.625\%$ of the toilet paper is left.