Q70 / 125 · BITSAT 2014 · Mathematics
The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and costs ' 48 per hour at 16 miles per hour. The most economical speed if the fixed charges i.e. salaries etc. amount to 300 per hour is
- A10
- B20
- C30
- Correct answerD40
Solution
Let the speed of the train be $\mathrm{v}$ and distance to be covered be so that total time taken is $\mathrm{s} / \mathrm{v}$ hours. Cost of fuel per hour $=\mathrm{kv}^{2}(\mathrm{k}$ is constant) Also $48=\mathrm{k} .16^{2}$ by given condition $\therefore \mathrm{k}=\frac{3}{16}$
$\therefore$ Cost to fuel per hour $\frac{3}{16} \mathrm{v}^{2}$. Other charges per hour are 300 . Total running cost,
$$
\begin{array}{r}
C=\left(\frac{3}{16} \mathrm{v}^{2}+300\right) \frac{\mathrm{s}}{\mathrm{v}}=\frac{3 \mathrm{~s}}{16} \mathrm{v}+\frac{300 \mathrm{~s}}{\mathrm{v}} \\
\frac{\mathrm{dC}}{\mathrm{dv}}=\frac{3 \mathrm{~s}}{16}-\frac{300 \mathrm{~s}}{\mathrm{v}^{2}}=0 \Rightarrow \mathrm{v}=40
\end{array}
$$
$\frac{\mathrm{d}^{2} \mathrm{C}}{\mathrm{dv}^{2}}=\frac{600 \mathrm{~s}}{\mathrm{v}^{3}}>0 \therefore \mathrm{v}=40$ results in minimum
running cost
Hence, the correct answer is option: 'D'.