I was shopping for some toilet paper, I saw an advertisment for double the amount of toilet paper as a single roll. Assuming that the toilet paper is equally densely packed, and that the diameter of the core is the same at 1 unit, and the diameter of the original roll is 5 units, what is the diameter of the new roll?
That means that instead of the (2.5)^2 π area, we have...
So, 2.5^2 = 6.25
minus (0.5)^2 π = 0.25π for the area of the middle
12π+0.25π = 12.25π
sqrt(12.25) = 3.5
7 units diameter
Oops I made a mistake earlier, Corrected now
The original roll has a diameter of 5 unts ---> radius = 2.5 units.
It's total area is pi·2.52 = 6.25pi units2.
However, not all of that is paper; there is a hollow core of diameter 1 unit ---> radius = 0.5 units.
The area of the core is pi·0.52 = 0.25pi units2.
Subtracting the area of the core from the total area gives 6.25pi units2 - 0.25pi units2 = 6.00pi units2.
To get twice the amount of paper: 2 x 6.00pi units2 = 12.00pi units2.
However, we can't forget the core, so we need 12.00pi units2 + 0.25pi units2 = 12.25pi units.
Finding the square root of 12.25, we get 3.5, so the roll needs a radius of 3.5 units,
which is a diameter of 7 units.