Question

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?

• A. 10
• B. 12
• C. 15
• D. 18

Hint:

When dealing with relative motion and distances, use the Pythagorean theorem to find the exact distance when two objects move at right angles.

Answer 1

The Correct Option is C

Step 1: Analyze car positions at 11:00 AM.
Car P travels at 25 km/h for 1.5 hours (10 AM to 11:30 AM), covering a distance of: \[ \text{Distance of car P} = 25 \times 1.5 = 37.5 \, \text{km}. \] Car Q travels at 30 km/h for 1 hour, covering a distance of: \[ \text{Distance of car Q in 1 hour} = 30 \times 1 = 30 \, \text{km}. \] Step 2: Apply the Pythagorean theorem.
At 11:30 AM, both cars are equidistant from X, forming a right triangle with X. To find the distance Q must have traveled to match P's position: \[ \text{Distance of car Q at 11:30 AM} = \sqrt{(30^2 + 37.5^2)} \approx 47.43 \, \text{km}. \] Since Q has only traveled 30 km in its first hour, it must stop to cover the remaining distance.
Step 3: Calculate Q's stopping time.
Car Q needs to cover an additional \( 47.43 - 30 = 17.43 \, \text{km} \). At 30 km/h, the time required is: \[ \text{Time taken to travel remaining distance} = \frac{17.43}{30} \times 60 = 34.86 \, \text{minutes}. \] Therefore, car Q stopped for approximately \( 34.86 - 15 = 15 \) minutes.