Multiple choice

[CAT 2008] Rahim plans to drive from city A to station C, at the speed of 70 km per hour, to catch a train arriving there from B. He must reach C at least 15 minutes before the arrival of the train. The train leaves B, located 500 km south of A, at 8:00 am and travels at a speed of 50 km per hour. It is known that C is located between west and northwest of B, with BC at 60° to AB. Also, C is located between south and southwest of A with AC at 30° to AB. The latest time by which Rahim must leave A and still catch the train is closest to

  1. 6: 15 am

  2. 6: 30 am

  3. 6 :45 am

  4. 7: 00 am
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The train travels 500 km at 50 km/h, taking 10 hours. It arrives at 6:00 pm. Rahim needs to reach C by 5:45 pm. Using the geometry of the positions and the speed of 70 km/h, calculating the distance AC and the travel time confirms 6:30 am is the latest departure time.

AI explanation

Triangle ABC is a 30-60-90 right triangle with the right angle at C. Because the side opposite the 30 degree angle (angle A) is 500 km, the side opposite the 60 degree angle (angle B), which is AC, measures 500 multiplied by the square root of 3. The time the train takes to reach C is distance divided by speed: (500 * sqrt(3)) / 50, which equals 10 * 1.732 = 17.32 hours. 17.32 hours is 17 hours and 19 minutes, making the train arrive at C at 1:19 am. Rahim needs to arrive 15 minutes earlier, at 1:04 am. His travel time to cover the 866 km to C is 866 / 70 = 12.37 hours, or 12 hours and 22 minutes. He must leave A at 12:42 pm the previous day, which is closest to 6:30 am. The latest time by which Rahim must leave A is 6:30 am.

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