# Relation between volume, diameter and height of right circular cone

The volumes of two right circular cones are in the ratio of 1:3 and their diameters are in the ratio of 3:5. The ratio of their height is

- 3 : 5
- 25 : 27
- 9 : 20
- 2 : 5

Answer: 25 : 27

**Solution:**

Volume of right circular cone, V = (1/3) πr^{2}h

Let us assume,

Volume, height and radius for first right circular cone are : V_{1}, h_{1} and r_{1} and V_{2}, h_{2} and r_{2} for second cone.

According to question,

3V_{1} = V_{2} and

5r_{1} = 3r_{2
}

From eqn. 1

3V_{1} = V_{2}

3 (1/3) π r_{1}^{2 }h_{1} = (1/3) π r_{2}^{2 }h_{2}

after solving this equation on both side

r_{1}^{2 }h_{1} = (1/3) r_{2}^{2 }h_{2}

then put value from eqn. 2

((3/5) r_{2 })^{2} h_{1} = (1/3) r_{2}^{2 }h_{2}

(9/25) r_{2}^{2} h_{1} = (1/3) r_{2}^{2 }h_{2}

h_{1} = 1/3 * 25/9 h_{2} —- r_{2}^{2} was canceled on both side

h_{1} = (25/27) h_{2
}

Final Answer: Ratio of height of cone1 and cone2 is = 25 : 27

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