Question

The diameter and height of a cylinder are measured by a meter scale to be $12.6\pm 0.1$ cm and $34.2\pm 0.1$ cm ,respectively. What will be the value of its volume in appropriate significant figures?

• A. $4264\pm 81{cm}^3$
• B. $4260\pm 80{cm}^3$
• C. $4264.4\pm 81.0{cm}^3$
• D. $4300\pm 80{cm}^3$

Answer 1

Answer: (B)

$4260\pm 80{cm}^3$

$\begin{array}{cc}\text{Given}\to & \text{diameter}\left(d\right)=12.6\pm 0.1cm\\ & \text{height}\left(h\right)=34.2\pm 0.1cm\end{array}$

$\text{Volume}=\pi \frac{d^2}{4}h=\frac{22}{7}\times \frac{12.6\times 12.6\times 34.2}{4}=4260{cm}^3$

$\begin{array}{c}\text{Taking log both side}\\ \log V=\log \left(\pi \frac{d^2}{4}h\right)\\ \log V=\log \pi +2\log d-\log 4+\mathrm{logh}\\ \text{Differentiate both side}\\ \frac{\delta V}{V}=0+2\frac{\delta d}{d}-0+\frac{\delta h}{h}\end{array}$

$\begin{array}{cc}\delta V& =(\frac{2\times 0.1}{12.6}+\frac{0.1}{34.2})V\\ & =(\frac{0.2}{12.6}+\frac{0.1}{34.2})4260\\ & =8{cm}^3\\ \therefore \text{Volume}& =4260\pm 80{cm}^3\end{array}$