SOLUTION: 1. What happens to the capacity of a bin if a) its height is doubled b) its height is trebled c) all of its dimensions are doubled d) all of its dimensions are trebled

Question 1037527: 1. What happens to the capacity of a bin if
a) its height is doubled
b) its height is trebled
c) all of its dimensions are doubled
d) all of its dimensions are trebled
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
let's assume the bin is a cylinder.
the volume of a cylinder is pi * r^2 * h
if the height is doubled, then the volume is pi * r^2 * h * 2
the ratio of the new volume to the old volume is r^2 * h * 2 divided by r^2 * h.
the r^2 and the h cancel out and you are left with 2.
if you double the height, you double the volume.
if you triple the height, you triple the volume.

let's now assume all the dimensions are doubled.
the volume is pi * r^2 * h
double all of the dimensions and you get pi * (2 * r)^2 * 2 * h
simplify to get pi * 4 * r^2 * 2 * h
simplify further to get 8 * r^2 * h
the volume is multiplied by 8 which is the same as multiplying by 2^3, or the cube of the multiple of the dimensions.
if you triple the dimensions, then the volume is multiplied by 3^3 = 27 times.

if the bin is rectangular, the same concepts apply.
double the height and the volume is doubled.
triple the height and the volume is tripled.
double all the dimensions and the volume is multiplied by 2^3.
triple all the dimensions and the volume is multiplied by 3^3.

for example:
volume is length * width * height.
double the height and the volume is length * width * height * 2
ratio of new volume to old volume is 2 to 1.
triple the height and hte volume is length * width * height * 3
ratio of new volume to old volume is 3 to 1.
double all the dimensions and the volume is length * 2 * width * 2 * height * 2.
ratio of new volume to old volume is 8 to 1 or 2^3 to 1.
triple all the dimensions and the volume is length * 3 * width * 3 * height * 2.
ratio of new volume to old volume is 27 to 1 or 3^3 to 1.

the general rule is:

multiplying only one of the dimensions by a factor of x leads to multiplying the volume by x.

multiplying only two of the dimensions by a factor of x leads to multiplying the volume by x^2.

multiplying all three of the dimensi0ons by a factor of x leads to multiplying the volume by x^3.

this can easily be seen by applying the formula to the rectangle.

length * width * height = 1 times.
2 * length * width * height = 2 times
2 * length * 2 * width * height = 4 times = 2^2 times.
2 * length * 2 * width * 2 * height = 8 times = 2^3 times.

with the cylinder, the same concept applies.

pi * r^2 * h = 1 time.
pi * r^2 * 2 * h = 2 times.
pi * (2 * r)^2 * h = 4 * r^2 * h = 4 times = 2^2 times
pi * (2 * r)^2 * 2 * h = 4 * r^2 * 2 * h = 8 * r^2 * h = 8 times = 2^3 times.

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