Question

Assuming that the petrol burnt in a motor boat varies as the cube of its velocity, the most economical speed, when going against a current of c km/h is

• A. (3c/2 ) km/h
• B. (3c/4) km/h
• C. (5c/2) km/h
• D. (c/2) km/h

Answer 1

The correct option is A

(3c/2 ) km/h

Suppose p is the quantity of petrolburnt in one hour. Then $p=k{v}^3$. Let the distance of the journey be s km.
Duration of the journey $=\frac{s}{v-c}$ hours
$\therefore$ Total petrol burnt for the whole journey
$=\frac{sp}{v-c}=\frac{sk{v}^3}{v-c}$
Take $u=\frac{v^3}{v-c}(sk=constant),$
$\frac{du}{dv}={\frac{(v-c)3v^2-v^3}{v-c}}^2=\frac{v^2(2v-3c)}{(v-c{)}^2}\frac{du}{dv}=0\Rightarrow v=\frac{3c}{2},\frac{3c}{2}isfoundtogiveaminimumforu.$
$\therefore$ Most economical speed $=\frac{3c}{2}$ km/h