QQuestion
The height of a right circular cone is twice the radius of the base. If the height were seven times the diameter of the base then the volume of the cone would have been 539 cu cm more. Find the height of the cone.
WBBSEClass XMathematicsRight Circular Cone04 Mark2023 Moderate
✓Answer
Let, radius of cylinder = r and the height of cylinder = 2r
If its height be 7 times its diameter, new height of cylinder = 14r
Case – 1: Volume = 1\over 3 2πr²h
= 1\over 3 × 2 × 22\over 7 × r² × 2r
= 44r³\over 21
Case – 2: Radius = r, Height = 14r
Volume = 1\over 3 πr²h
= 1\over 3 × 2 × 22\over 7 × r² × 14r
= 308r³\over 21
According to the Question,
14\over 3 × πr³ – 2\over 3 × πr³ = 539
or, 4πr³ = 539
or, 4 × 22\over 7 × r³ = 539
or, r³ = 539 × 7\over 22 × 4
or, r³ = 7 × 7 × 7 \over 2 × 2 × 2
or, r = 7\over 2 = 3.5 cm
Given Height is twice of the radius,
H = r × 2 = 3.5 × 2 = 7 cm
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