Question

If the petrol burnt per hour in driving a motor boat varies as the cube of its velocity when going against a current of C kmph, the most economical speed is

• A. $\frac{c}{2}$
• B. $\frac{3c}{2}$
• C. $\frac{\sqrt{3}c}{2}$
• D. $c$

Answer 1

Answer: (B)

$\frac{3c}{2}$

Let the petrol burnt per hour be 'y' units and V kmph be the velocity of the motor boat..
Since, $y\propto V^3$
$\Rightarrow y=k{V}^3$ where 'k' is proportionality constant.
Velocity of boat against current $=$ (V-C)kmph.
Let S be the distance travelled by the boat in time t.
Then petrol burnt $=\frac{S}{V-C}\times k{V}^3$
Let $f\left(V\right)=\frac{S}{V-C}\times k{V}^3$
$f^{\prime }\left(V\right)=\frac{2V^3-3V^{2}C}{(V-C{)}^2}$
For maxima or minima,
$f^{\prime }\left(V\right)=0$
$\Rightarrow V=0,\frac{3C}{2}$
Clearly $f^{\prime \prime }\left(V\right)>0$ for $V=\frac{3C}{2}$
Hence, most economical speed is $\frac{3C}{2}$