Respuesta :
Answer: 15.625%
===============================
Work Shown:
Let's compute the volume of the entire cylinder. For now it includes the empty space.
d = 12 is the diameter
r = d/2 = 12/2 = 6 is the radius
V = pi*r^2*h = pi*6^2*h = 36pi*h is the volume.
Let A = 36pi*h so we can use it later.
--------
Now let's compute the volume of the empty space
d = 4 is the diameter of the empty space cylinder
r = d/2 = 4/2 = 2 is the radius
V = pi*r^2*h = pi*2^2*h = 4pi*h is the volume
Let B = 4pi*h
-------
Subtract the two results: A-B = 36pi*h - 4pi*h = (36-4)*pi*h = 32pi*h
The volume of just the toilet paper, without the empty space, is exactly 32pi*h
Let C = 32pi*h
--------
Repeat all of those steps done in the prior first and third sections; however, this time the starting diameter is now d = 6 (instead of d = 12). The empty space will not change so we don't have to recompute the value of B.
E = volume of entire cylinder with diameter 6 (radius 3)
E = pi*r^2*h
E = pi*3^2*h
E = 9pi*h
F = volume of toilet paper left
F = (entire cylinder with empty space) - (empty space)
F = E - B
F = 9pi*h - 4pi*h
F = 5pi*h
The toilet paper roll that has diameter 6 cm has a volume of 5pi*h cubic cm. This is ignoring the empty space.
--------
We have
C = 32pi*h
F = 5pi*h
as the starting and ending volumes of the toilet paper.
Compute the percent change
P = percent change
P = 100*(new - old)/(old)
P = 100*(F - C)/C
P = 100*(5pi*h - 32pi*h)/(32pi*h)
P = 100pi*h(5-32)/(32pi*h)
P = 100pi*h*(-27)/(32pi*h)
P = 100*(-27)/(32)
P = -84.375
note how the 'h' terms cancel out, so the height does not affect the final answer. The 'pi' terms cancel as well.
The percent change is -84.375% meaning that the starting roll lost 84.375% of its volume to drop to the current roll of toilet paper.
Subtract that from 100% to figure out what percentage is left over
100% - 84.375% = 15.625%
