Question

The volume of a cylinder is computed from measurements of its height (h) and diameter (d). A set of several measurements of height has an average value of 0.2 m and a standard deviation of 1%. The average value obtained for the diameter is 0.1 m and the standard deviation is 1%. Assuming the errors in the measurements of height and diameter are uncorrelated, the standard deviation of the computed volume is

• A. 1.00%
• B. 1.73%
• C. 2.23%
• D. 2.41%

Answer 1

The correct option is C 2.23%

Volume of cylinder,

$V=\pi r^{2}h$

$\Rightarrow \frac{\partial V}{\partial r}=2\pi rh$

$\frac{\partial V}{\partial h}=\pi r^2$

Given, h = 0.2 m and d = 0.1 m

$\Rightarrow r=0.05m;{\sigma }_r=\frac{1}{100}\times 0.05=5\times {10}^{-4}$

${\sigma }_h=0.2\times \frac{1}{100}=2\times {10}^{-3}$

Standard deviation of volume,

${\sigma }_h=0.2\times \frac{1}{100}=2\times {10}^{-3}$

$\Rightarrow {\sigma }_v=\sqrt{(2\pi rh{)}^2.(5\times {10}^{-4}{)}^2+\left(\pi r^2{)}^2\right(2\times {10}^{-3}{)}^2}$

$\Rightarrow {\sigma }_v=3.512\times {10}^{-4}$

$\Rightarrow \%{\sigma }_v=\frac{3.5\times {10}^{-4}}{\pi \times (0.05{)}^2\times 0.2}\times 100\%=2.23\%$