For a steamer,the consumption of petrol (per | vedclass

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(B) Let $V$ be the speed of the steamer in still water and $d$ be the distance to be covered.The speed of the steamer against the current is $(V - C)$

Vedclass · Accepted Answer

$1.5$

Vedclass · Answer

(B) Let $V$ be the speed of the steamer in still water and $d$ be the distance to be covered.The speed of the steamer against the current is $(V - C)$.The time taken for the journey is $T = \frac{d}{V - C}$.The petrol consumption per hour is given by $P = kV^3$,where $k$ is a constant.The total fuel consumption $F$ is given by $F = T 	imes P = \frac{d}{V - C} 	imes kV^3 = kd \frac{V^3}{V - C}$.To find the most economical speed,we minimize $F$ with respect to $V$ by setting $\frac{dF}{dV} = 0$.Let $f(V) = \frac{V^3}{V - C}$.Using the quotient rule: $f'(V) = \frac{(V - C)(3V^2) - V^3(1)}{(V - C)^2} = 0$.This implies $3V^2(V - C) - V^3 = 0$.$3V^3 - 3V^2C - V^3 = 0$.$2V^3 - 3V^2C = 0$.Since $V 
eq 0$,we divide by $V^2$: $2V - 3C = 0$.$V = 1.5C$.Thus,the most economical speed of the steamer in still water is $1.5C$.

For a steamer,the consumption of petrol (per hour) varies as the cube of its speed (in $km/hr$). If the speed of the current is steady at $C \, km/hr$,then the most economical speed of the steamer going against the current will be ........... $C$.

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