Question Bank • QB_EM_100 NAT • 2 Marks

Section 1: Engineering Mathematics

Calculus

Subtopic: Total differentiation

Question Statement:

A right circular cone has radius $r = 10\text{ cm}$ and height $h = 20\text{ cm}$. If radius increases by $1\%$ and height decreases by $1\%$, calculate the approximate percentage change in its volume.

Answer Key:

1

Step-by-Step Solution & Explanation

Volume of a cone $V = \frac{1}{3} \pi r^2 h$. Taking natural logarithm: $\ln V = \ln\left(\frac{\pi}{3}\right) + 2 \ln r + \ln h$ Differentiating totally: $\frac{dV}{V} = 2 \frac{dr}{r} + \frac{dh}{h}$ Given $\frac{dr}{r} = +1\% = +0.01$ and $\frac{dh}{h} = -1\% = -0.01$: $\frac{dV}{V} = 2(+1\%) + (-1\%) = +2\% - 1\% = +1.0\%$
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