Question

The error in the measurement of the radius of a sphere is $0.5\%$. What is the permissible percentage error in the measurement of its volume?

• A. $15\%$
• B. $2.5\%$
• C. $3.5\%$
• D. $1.5\%$

Answer 1

The correct option is D $1.5\%$

$\text{Volume}\left(V\right)=\frac{4}{3}\pi r^3$

$\begin{array}{c}\text{Taking log.}\\ \text{log(V)}=\log \left(\frac{y}{3}\pi \right)+3\log r\\ \text{Differentiating}\\ \frac{\delta V}{V}=3\frac{\delta r}{r}\\ \frac{\delta V}{V}\times \mathrm{100}=3\frac{\delta r}{r}\times 100\dots \left(i\right)\end{array}$

$\begin{array}{c}\frac{\delta r}{r}\times 100=0.5\%\\ \text{putting in (i)}\\ \begin{array}{cc}\frac{\delta V}{V}\times 100& =3\times 0.5\\ & =1.5\%\end{array}\end{array}$