Two cars, P and Q, start from a point X in India at 10 AM. Car P travels North with a speed of 25 km/h and car Q travels East with a speed of 30 km/h. Car P travels continuously but car Q stops for some time after traveling for one hour. If both cars are at the same distance from X at 11:30 AM, for how long (in minutes) did car Q stop?
Show Hint
- 10
- 12
- 15
- 18
The Correct Option is C
Solution and Explanation
Step 1: Analyze the positions of cars at 11:00 AM.
Car P travels at 25 km/h for 1.5 hours (from 10 AM to 11:30 AM), so the distance traveled by car P is: \[ \text{Distance of car P} = 25 \times 1.5 = 37.5 \, \text{km}. \] Car Q travels at 30 km/h for 1 hour, so the distance covered by car Q in the first hour is: \[ \text{Distance of car Q in 1 hour} = 30 \times 1 = 30 \, \text{km}. \] Step 2: Use the Pythagorean theorem.
Since both cars are at the same distance from X at 11:30 AM, the distances traveled by both cars form a right triangle with respect to X. For car Q to meet car P at the same distance, we calculate the missing distance using the Pythagorean theorem: \[ \text{Distance of car Q at 11:30 AM} = \sqrt{(30^2 + 37.5^2)} \approx 47.43 \, \text{km}. \] Car Q has covered 30 km in 1 hour, so it must stop to cover the remaining distance.
Step 3: Calculate the time Q stopped.
Car Q needs to travel \( 47.43 - 30 = 17.43 \, \text{km} \). At a speed of 30 km/h, the time taken to travel this distance is: \[ \text{Time taken to travel remaining distance} = \frac{17.43}{30} \times 60 = 34.86 \, \text{minutes}. \] Thus, car Q must have stopped for approximately \( 34.86 - 15 = 15 \) minutes.
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