Q111 / 160 · AP EAMCET 2018 (22 Apr Shift 2) · Mathematics

If the petrol burnt in driving a motor boat varies as the cube of the velocity, then the speed (in $\mathrm{km} /$ hour) of the boat going against a water flow of $C \mathrm{kms} /$ hour so that the quantity of petrol burnt is minimum is

  • A
    $\frac{2 C}{3}$
  • Correct answer
    B
    $\frac{3 C}{2}$
  • C
    $\frac{4 C}{3}$
  • D
    $\frac{3 C}{4}$

Solution

Let $V>C$ the speed of motor boat, relative to water. In fact, relative to land, the motor boat moves with the speed $V-C$. The motor boat covers a distance $S$ in $\frac{S}{V-C} \mathrm{~h}$ and consumption is $f(v)=k v^3 \cdot \frac{S}{V-C}$, where $k$ is a constant. The minimum of consumption is attained for $$ \begin{aligned} & f^{\prime}(v)=0 \\ & \Rightarrow \quad k S \frac{(v-C) 3 v^2-v^3(1)}{(v-C)^2}=0 \\ & \Rightarrow \quad 3 v^3-3 v^2 C-v^3=0 \\ & \Rightarrow \quad 2 v^3-3 v^2 C=0 \Rightarrow v=\frac{3 C}{2} \text {. } \\ & \end{aligned} $$

Hence, the correct answer is option: 'B'.

Relevant Questions