If the petrol burnt in driving a motor boat v | vedclass

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Question

(B) Let $v$ be the speed of the motor boat relative to the water,where $v > C$. The speed of the boat relative to the land,while moving against the wa

Vedclass · Accepted Answer

$\frac{3C}{2}$

Vedclass · Answer

(B) Let $v$ be the speed of the motor boat relative to the water,where $v > C$. The speed of the boat relative to the land,while moving against the water flow,is $(v - C)$.If the distance to be covered is $S$,the time taken is $t = \frac{S}{v - C}$.The rate of petrol consumption per hour is proportional to the cube of the velocity relative to the water,i.e.,$k v^3$,where $k$ is a constant.The total petrol consumed $f(v)$ is given by:$f(v) = (k v^3) 	imes \left( \frac{S}{v - C} ight) = k S \frac{v^3}{v - C}$.To find the minimum consumption,we differentiate $f(v)$ with respect to $v$ and set it to zero:$f'(v) = k S \left[ \frac{(v - C)(3v^2) - v^3(1)}{(v - C)^2} ight] = 0$.This implies $3v^2(v - C) - v^3 = 0$,which simplifies to $3v^3 - 3v^2C - v^3 = 0$.$2v^3 - 3v^2C = 0$.Since $v 
eq 0$,we have $2v - 3C = 0$,which gives $v = \frac{3C}{2}$.

If the petrol burnt in driving a motor boat varies as the cube of the velocity,then the speed (in $km/h$) of the boat relative to the water,going against a water flow of $C \ km/h$,so that the quantity of petrol burnt is minimum,is:

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