Q.3 The radius as well as the height of a circular cone increases by 10%. The percentage
increase in its volume is ______.
(A) 17.1 (B) 21.0 (C) 33.1 (D) 72.8
Percentage Increase in Volume of Circular Cone: Radius and Height Up 10%
The volume of a circular cone increases by 33.1% when both its radius and height increase by 10%. This matches option (C).
Solution Steps
The volume V of a cone is:
V = (1/3)πr²h
New radius = 1.1r, new height = 1.1h.
New volume:
V′ = (1/3)π(1.1r)²(1.1h)
= (1/3)πr²h × (1.1)³
= 1.331V
Percentage increase:
[(V′ − V) / V] × 100% = (1.331 − 1) × 100% = 33.1%
Option Analysis
- (A) 17.1: Incorrect; ignores r² factor.
- (B) 21.0: Incorrect; corresponds to surface area increase (1.1² = 1.21).
- (C) 33.1: Correct, from (1.1)³ = 1.331
- (D) 72.8: Incorrect; this is for 20% increase (1.2)³ = 1.728.
Introduction
In competitive exams like IIT JAM, questions on percentage increase in volume of circular cone when radius and height increase by 10% test mensuration basics. The key phrase percentage increase in volume of circular cone highlights this vital concept, where volume scales with r²h, leading to a 33.1% rise—not simple addition of percentages.
Volume Formula Breakdown
The cone volume formula V = (1/3)πr²h shows dependency on radius squared and height linearly. A 10% rise multiplies radius by 1.1 (so r² becomes 1.21) and height by 1.1, yielding (1.1)³ = 1.331 times original volume.
Step-by-Step Calculation
Original: V = (1/3)πr²h
New: V′ = (1/3)π(1.1r)²(1.1h) = 1.331V
Increase: 33.1%
Common Exam Traps
Students pick 21% (confusing with lateral surface) or 17.1% (additive error). Always compute (1 + p/100)³ − 1 for p=10%. For 20%, it’s 72.8%.
Exam Tips
Practice with IIT JAM past papers; remember scaling factors for similar figures.
