Question : The heights of two right circular cones are in the ratio 1 : 5 and the perimeter of their bases are in the ratio 5 : 3. Find the ratio of their volumes.
Option 1: 8 : 11
Option 2: 7 : 6
Option 3: 5 : 9
Option 4: 3 : 4
Correct Answer: 5 : 9
Solution :
The volume of a cone, where $r$ is the radius and $h$ is the height.
$V = \frac{1}{3}\pi r^2 h$
Given that the heights of the cones are in the ratio 1 : 5.
Let the height of the first cone be $h$ and the height of the second cone be $5h$.
The perimeter of the base of a cone = $2\pi r$, which is in the ratio 5 : 3.
Let the radius of the first cone be $5r$ and the radius of the second cone be $3r$.
$⇒V_1 = \frac{1}{3}\pi (5r)^2 h = \frac{25}{3}\pi r^2 h$
$⇒V_2 = \frac{1}{3}\pi (3r)^2 5h = \frac{45}{3}\pi r^2 h$
The ratio of their volumes,
$⇒ V_1:V_2 = \frac{25}{3}\pi r^2 h:\frac{45}{3}\pi r^2 h = 5:9$
Hence, the correct answer is 5 : 9.
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