Topic: Engineering Mathematics
If a quantity Q is dependent on three other quantities $q_{1}$, $q_{2}$ and $q_{3}$ related such that $Q=K\times(q_{1})^{n_{1}}\times(q_{2})^{n_{2}}\times(q_{3})^{n_{3}}$ then overall error $\frac{\delta Q}{Q}=$
- A. $n_{1}(\frac{\delta q_{1}}{q_{1}})+n_{2}(\frac{\delta q_{2}}{q_{2}})+n_{3}(\frac{\delta q_{3}}{q_{3}})$
- B. $\frac{1}{n_{1}}\frac{\delta q_{1}}{q_{1}}+\frac{1}{n_{2}}\frac{\delta q_{2}}{q_{2}}+\frac{1}{n_{3}}\frac{\delta q_{3}}{q_{3}}$
- C. $\frac{\delta q_{1}}{q_{1}}+\frac{\delta q_{2}}{q_{2}}+\frac{\delta q_{3}}{q_{3}}$
- D. $(\frac{\delta q_{1}}{q_{1}})^{n_{1}}+(\frac{\delta q_{2}}{q_{2}})^{n_{2}}+(\frac{\delta q_{3}}{q_{3}})^{n_{3}}$
Answer: A. $n_{1}(\frac{\delta q_{1}}{q_{1}})+n_{2}(\frac{\delta q_{2}}{q_{2}})+n_{3}(\frac{\delta q_{3}}{q_{3}})$
Explanation: $Q=K\times(q_{1})^{n_{1}}\times(q_{2})^{n_{2}}\times(q_{3})^{n_{3}}$. Taking log both side, $\ln Q=\ln k+n_{1} \ln q_{1}+n_{2} \ln q_{2}+n_{3} \ln q_{3}$. Differentiating both side: $\frac{\delta Q}{Q}=n_{1}(\frac{\delta q_{1}}{q_{1}})+n_{2}(\frac{\delta q_{2}}{q_{2}})+n_{3}(\frac{\delta q_{3}}{q_{3}})$. $\ln k=0$, because 'k' is constant.