SOLUTION: Two cylinders are such that the ratio of their base radii is 2 : 1 and the ratio of their heights is 3 : 1. Find the ratio of their respective volumes.

Question 967595: Two cylinders are such that the ratio of their base radii is 2 : 1 and the ratio of their heights is 3 : 1. Find the ratio of their respective volumes.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
ratio of heights is 3 to 1.
ratio of radii of the base is 2 to 1.

ratio of volume is?

volume = pi * r^2 * h

let the radius of the smaller cylinder be r.
then the radius of the larger cylinder is 2r.

let the height of the smaller cylinder be h.
then the height of the larger cylinder is 3h.

the volume of the smaller cylinder is pi * r^2 * h.

the volume of the larger cylindere is pi * (2r)^2 * (3h).

let v1 = volume of smaller cylinder.

let v2 = volume of larger cylinder.

the ratio of the volume of the smaller cylinder to the volume of the larger cylinder is:

v1 / v2 = (pi * r^2 * h) / (pi * (2 * r)^2 * (3 * h)

simplify to get:

v1 / v2 = (pi * r^2 * h) / (pi * 4 * r^2 * 3 * h)

the pi and the r^2 and the h in the numerator and denominator cancel out and you are left with:

v1 / v2 = 1 / (4 * 3) which simplfies to:

v1 / v2 = 1 / 12.

that'a the ratio of the volumes.

as an example:

let the radius of the smaller cylinder be 5 and the height of the smaller cylinder be 10.

the radius of the larger cylinder is 10 and the height of the larger cylinder is 30.

v1 = pi * 5^2 * 10 = 250 * pi.

v2 = pi * 10^2 * 30 = 3000 * pi.

v1 / v2 = 250 / 3000 - 25 / 300 = 1 / 12.

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