Two agricultural drones, Drone A and Drone B, start simultaneously from the same point on a circular field boundary of perimeter $1,200\text{ m}$ and fly in opposite directions with uniform ground speeds of $18\text{ m/s}$ and $12\text{ m/s}$, respectively. After how many seconds from the start will they meet for the first time at the EXACT starting point?
Options:
(A) $200\text{ seconds}$
(B) $100\text{ seconds}$
(C) $40\text{ seconds}$
(D) $240\text{ seconds}$
Answer Key:
A
Step-by-Step Solution & Explanation
**Method 1: Individual Lap Periodicities and LCM**
- Time for Drone A to complete one full round: $T_A = \frac{1200}{18} = \frac{200}{3}\text{ s}$.
- Time for Drone B to complete one full round: $T_B = \frac{1200}{12} = 100\text{ s}$.
They meet at the starting point at a time $T = \text{LCM}(T_A, T_B)$:
$T = \text{LCM}\left(\frac{200}{3}, \frac{100}{1}\right) = \frac{\text{LCM}(200, 100)}{\gcd(3, 1)} = \frac{200}{1} = 200\text{ seconds}$
At $t = 200\text{ s}$, Drone A has completed $200 \times 18 / 1200 = 3$ laps, and Drone B has completed $200 \times 12 / 1200 = 2$ laps. Option **A** is correct.