Multiple choice

Two trains leave a railway station at the same time. The first train travels due west and the second train due north. The first train travels $5$ km/hr faster than the second train. If after two hours, they are $50$ km apart, find the average speed of each train.

  1. Speed of first train $=$ $20$ km/hr , speed of second train $=$ $15$ km/hr
  2. Speed of first train $=$ $25$ km/hr , speed of second train $=$ $20$ km/hr
  3. Speed of first train $=$ $15$ km/hr , speed of second train $=$ $10$ km/hr
  4. Speed of first train $ = $ $30$ km/hr , speed of second train $=$ $25$ km/hr
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A Correct answer
Explanation

Let speed of second train be v. First train is v+5. After 2 hours, distances are 2v and 2(v+5). By Pythagoras, (2v)^2 + (2(v+5))^2 = 50^2. 4v^2 + 4(v^2+10v+25) = 2500. 8v^2 + 40v + 100 = 2500. 8v^2 + 40v - 2400 = 0. v^2 + 5v - 300 = 0. (v+20)(v-15) = 0. v=15. First train = 20, second = 15.

AI explanation

Let the speed of the second train be x km/hr, making the speed of the first train x plus 5 km/hr. Using the Pythagorean theorem for the distances covered in 2 hours, 2 times x plus 5 squared plus 2x squared equals 50 squared. Expanding gives 4x squared plus 20x plus 25 plus 4x squared equals 2500, which simplifies to x squared plus 2.5x minus 306.25 equals 0. Solving this quadratic equation gives x equals 15 km/hr, meaning the first train is 20 km/hr.

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