Question

The cost of running a bus from $A$ to $B$ , is $Rs.\left(av+\frac{b}{v}\right)$ where $v$ km/h is the average speed of the bus. When the bus travels at $30\, km/h$, the cost comes out to be $Rs.\, 75$ while at $40\, km/h$, it is $Rs. \,65$. Then the most economical speed (in $km/ h$) of the bus is :

• A. $45$
• B. $50$
• C. $60$
• D. $40$

Answer 1

Answer: (C)

$60$

Answer 2

Let cost $C=av+\frac{b}{v}$ According to given question, $30a+\frac{b}{30}=75\,...\left(i\right)$ $40a+\frac{b}{40}=65\,...\left(ii\right)$ On solving $\left(i\right)$ and $\left(ii\right)$, we get $a=\frac{1}{2} and b=1800$ Now, $C=av+\frac{b}{v}$ $\Rightarrow \frac{dC}{dv}=0 = a-\frac{b}{v^{2}}$ $\frac{dC}{dv}=0 \Rightarrow a-\frac{b}{v^{2}}=0$ $\Rightarrow v=\sqrt{\frac{b}{a}}=\sqrt{3600}$ $\Rightarrow v=60\,kmph$