Question

The height of a cylinder is equal to the radius. If an error of $\alpha$ % is made in the height, then percentage error in its volume is

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

Answer 1

Answer: (C)

$3\alpha$ %

Volume of cylinder $V=\pi r^{2}h$

Since, $h=r$

$V=\pi h^3$

$\frac{dV}{dh}=3\pi h^2$

Given, percentage error in measuring height $=\alpha$%

$\Rightarrow \frac{\Delta h}{h}=\frac{\alpha }{100}$

$\Rightarrow \Delta h=\frac{\alpha h}{100}$

Now, approximate error in measuring V$=dV=\left(\frac{dV}{dh}\right)\Delta h$

$=\frac{3\alpha }{100}\pi h^3=3\alpha$% of V

Percentage error in measuring $V=3\alpha$%