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Question:

Two straight roads R₁ and R₂ diverge from a point A at an angle of 120°. Ram starts walking from point A along R₁ at a uniform speed of 3 km/hr. Shyam starts walking at the same time from A along R₂ at a uniform speed of 2 km/hr. They continue walking for 4 hours along their respective roads and reach points B and C on R₁ and R₂, respectively. There is a straight line path connecting B and C. Then Ram returns to point A after walking along the line segments BC and CA. Shyam also returns to A after walking along line segments CB and BA. Their speeds remain unchanged. The time interval (in hours) between Ram’s and Shyam’s return to the point A is:

Explanation:

After 4 hours,

Distance travelled by Ram, AB = 4 × 3 = 12 km

Distance travelled by Shyam, AC = 4 × 2 = 8 km

Ram’s return path = BC + CA = (x + 8) km

Shyam’s return path = CB + BA = (x + 12) km

Now dropping perpendiculars from C and A to form the right triangle AOC,

In ΔAOC,

OA = AC sin30˚ = 8 × 1/2 = 4

OC = AC sin60° = 8 × $\frac{\sqrt{3}}{2} = 4 \sqrt{3}$

∴ BC = $\sqrt{({(4 \sqrt{3})}^{2} + 16^{2})}$

BC = $\sqrt{304}$

BC = $4 \sqrt{19}$

Time taken by Ram to travel = $\frac{B C + C A}{3} = \frac{4 \sqrt{19} + 8}{3}$ ...(i)

Time taken by Shyam to travel = $\frac{B C + B A}{2} = \frac{4 \sqrt{19} + 12}{2}$ ...(ii)

∴ Time interval between the two reaching point A is (i) subtracted from (ii)

= $\frac{2 \sqrt{19} + 10}{3}$

Hence, option (b).

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