The volume of two cones are in the ratio 1 : 4 and their diameters are 4 : 5,…
2013
The volume of two cones are in the ratio 1 : 4 and their diameters are 4 : 5, then the ratio of their height is
- A.
2 : 5
- B.
5 : 4
- C.
15 : 16
- D.
25 : 64
Show answer & explanation
Correct answer: D
A cone's volume is proportional to the square of its radius times its height: Volume proportional to radius squared times height. So for two cones being compared, Volume ratio = (radius ratio squared) times (height ratio).
Diameter and radius are directly proportional, so the radius ratio of the two cones equals their diameter ratio: 4 : 5.
From the concept relation, Volume ratio = (radius ratio squared) times (height ratio). The given volume ratio is 1 : 4, written as the fraction 1/4.
Substitute the radius ratio: 1/4 = (4/5 squared) times (height ratio) = (16/25) times (height ratio).
Solve for the height ratio: height ratio = (1/4) divided by (16/25) = (1/4) times (25/16) = 25/64.
So the ratio of the two heights is 25 : 64.
Check: with radius ratio 4 : 5 and height ratio 25 : 64, (radius ratio squared) times (height ratio) = (16/25) times (25/64) = 16/64 = 1/4, which reproduces the given volume ratio of 1 : 4, confirming the result.