The cost of running a bus from $A$ to $B$, is $₹\left(a v+\frac{b}{v}\right)$, where $v \mathrm{~km} /…
The cost of running a bus from $A$ to $B$, is $₹\left(a v+\frac{b}{v}\right)$, where $v \mathrm{~km} / \mathrm{h}$ is the average speed of the bus. When the bus travels at $30 \mathrm{~km} / \mathrm{h}$, the cost comes out to be $₹ 75$ while at $40 \mathrm{~km} / \mathrm{h}$, it is $₹ 65$. Then the most economical speed (in $\mathrm{km} / \mathrm{h}$ ) of the bus is :
45
50
60
40
Solution
Let $\operatorname{cost} \mathrm{C}=a v+\frac{b}{v}$
According to given question,
$
\begin{aligned}
& 30 a+\frac{b}{30}=75 \\
& 40 a+\frac{b}{40}=65
\end{aligned}
$
On solving (i) and (ii), we get $a=\frac{1}{2}$ and $b=1800$
Now, $C=a v+\frac{b}{v}$
$
\begin{aligned}
& \Rightarrow \frac{d C}{d v}=a-\frac{b}{v^2} \\
& \frac{d C}{d v}=0 \Rightarrow a-\frac{b}{v^2}=0 \\
& \Rightarrow v=\sqrt{\frac{b}{a}}=\sqrt{3600} \\
& \Rightarrow v=60 \mathrm{kmph}
\end{aligned}
$