Question 197 of 308

A man in a car at location Q on a straight highway is moving with speed $\mathrm{v}$. He decides to reach a point $P$ in a field at a distance $d$ from highway (point $M$ ) as shown in the figure.Speed of the car in the field is half to that on the highway. What should be the distance RM, so that the time taken to reach $P$ is minimum?

A$\frac{\mathrm{d}}{\sqrt{3}}$B$\frac{\mathrm{d}}{2}$C$\frac{\mathrm{d}}{\sqrt{2}}$D$\mathrm{d}$

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Answer: (A) $\frac{\mathrm{d}}{\sqrt{3}}$

Let the car turn of the highway at a distance ' $x$ ' from the point $\mathrm{M}$. So, $\mathrm{RM}=x$ And if speed of car in field is $v$, then time taken by the car to cover the distance $Q R=$ $Q M-x$ on the highway,
$$t_1=\frac{Q M-x}{2 v}$$
Time taken to travel the distance ' $R P^{\prime}$ in the field
$$t_2=\frac{\sqrt{d^2+x^2}}{v}$$
Total time elapsed to move the car from $Q$ to $P$
$$t=t_1+t_2=\frac{Q M-x}{2 v}+\frac{\sqrt{d^2+x^2}}{v}$$


$$\text { or } x=\frac{d}{\sqrt{2^2-1}}=\frac{d}{\sqrt{3}}$$