Question

If the ratio of base radius and height of a cone is 1:2 and percentage error in radius is $\lambda$ %, then the error in its volume is

• A. $\lambda$ %
• B. $2\lambda$%
• C. $3\lambda$%
• D. none of these

Answer 1

The correct option is A $\lambda$ %

Volume of cone $V=\frac{1}{3}\pi r^{2}h$

Given, $\frac{r}{h}=\frac{1}{2}$

$\Rightarrow V=\frac{2}{3}\pi r^3$

$\frac{dV}{dr}=2\pi r^2$

Percentage error in measuring r $=\lambda$%

$\Rightarrow \frac{\Delta r}{r}=\frac{\lambda }{100}$

$\Rightarrow \Delta r=\frac{\lambda r}{100}$

Approximate error in V $=dV=\left(\frac{dV}{dr}\right)\Delta r$

$=\frac{\lambda }{100}\left(2\pi r^3\right)=\lambda \%$ of V

Percentage error in $V=\lambda \%$