Question : The ratio of the radii of the two cones is 5 : 6 and their volumes are in the ratio of 8 : 9. The ratio of their heights is:
Option 1: 32 : 25
Option 2: 25 : 32
Option 3: 27 : 20
Option 4: 20 : 27
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Correct Answer: 32 : 25
Solution :
The volume of the cone = $\frac{1}{3}$πr
2
h, where r is the radius and h is the height.
Let the radius of one cone be 5r and 6r.
height is h
1
and h
2
⇒ $\frac{1}{3}$(π × 25r × h
1
) : $\frac{1}{3}$(π × 36r × h
2
) = 25h
1
: 36h
2
According to the question,
25h
1
: 36h
2
= 8 : 9
⇒ h
1
: h
2
= (8 × 36) : (25 × 9)
$\therefore$ h
1
: h
2
= 32 : 25
Hence, the correct answer is 32 : 25.
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