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 $30km/h$, the cost comes out to be $Rs.75$ while at $40km/h$, 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)

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
$a=\frac{1}{2}andb=1800$
Now, $C=av+\frac{b}{v}$
$\Rightarrow \frac{dC}{dv}=0=a-\frac{b}{v^2}$
$\frac{dC}{dv}=0\Rightarrow a-\frac{b}{v^2}=0$
$\Rightarrow v=\sqrt{\frac{b}{a}}=\sqrt{3600}$
$\Rightarrow v=60kmph$