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Question 885348: the volume of a cylinder is found using the formula v=πr2h, where r is the radius of the base and h is the height. The volume of a rectangular prism of the same width (2r) and height h, as the cylinder, is found by multiplying the area of the base (2rx2r) by the height. What is the ratio of the volume of the cylinder to the volume of the prism?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! volume of cylinder is equal to pi * r^2 * h
volume of rectangular prism is equal to l * w * h
l is the length
w is the width
h is the height
since w = 2 * r, the volume of the rectangular prism becomes equal to l * 2 * r * h.
let v1 = volume of the cylinder
let v2 = volume of the prism.
you get:
v1 = pi * r^2 * h
v2 = l * 2 * r * h
ratio of the cylinder to the prism is equal to v1/v2 which becomes:
v1/v2 = (pi * r^2 * h) / (l * 2 * r * h)
simplify this to get:
v1/v2 = (pi * r) / (2 * l)
r is the radius of the cylinder, l is the length of the prism.
that's your ratio.
you can confirm by making a test with some numbers.
assume r = 3 and h = 9
volume of the cylinder is equal to pi * r^2 * h = pi * 9 * 9 = pi * 81.
assume l = 5, w = 6, h = 9 for the prism.
the width of 6 is equal to 2 times the radius of 3.
volume of the prism is equal to 5 * 6 * 9 = 270
ratio of the volume of the cylinder to the prism is equal to (pi * 81) / 270.
divide numerator and denominator of this fraction by 27 and you get (pi * 3) / 10.
since 10 is equal to 2 * 5, the ratio becomes:
(pi * 3) / (2 * 5)
since 3 is the radius and 5 is the length, then this ratio is equivalent to:
(pi * r) / (2 * l) which confirms the formula for the ratio is correct.
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