While traveling at a speed of $x$ km/hr a bus burns petrol $(\frac{3}{x}+\frac{x}{300})$  litres/km. If cost of petrol be Rs 30/litre and other cost is also Rs 30/h then speed for a journey of 75 km having minimum cost is $20\sqrt{\lambda }$  km/hr. Find the value of  $\lambda$.

While traveling at a ……….
Let speed of bus is x km/hr
Time taken  $=\frac{75}{x}hr$
Other cost  $=30\times \frac{75}{x}$
Petrol cost  $=30\times [\frac{3}{x}+\frac{x}{300}]75$
Total cost C  $=30\frac{75}{x}+30[\frac{3}{x}+\frac{x}{300}]75$
 $=75[\frac{120}{x}+\frac{x}{10}]$
 $\frac{dc}{dx}=75[\frac{-120}{x^2}+\frac{1}{10}]$
For min cost  $\frac{dc}{dx}=0$
$x^2=1200\Rightarrow x=20\sqrt{3}$ 
$\frac{d^{2}c}{d{x}^2}=75\times \frac{240}{x}$ 
 $\frac{d^{2}c}{d{x}^2}>0atx=20\sqrt{3}$