Trains and distance?

Question

Two trains leave a station at the same time on straight sections of track which form an angle of 112 with each other. The first train, a freight train, travels at 52 mph and the second, a bullet passenger train, at 106 mph. How far apart are the trains, correct to the nearest 0.1 mile, after one and a half hours?

How do I solve this? Thank you.

Answer 1

After an hour and a half,
the first train has traveled 78 miles and
the second train has travelled 159 miles.
If x = distance between them, then by cosine rule,

cos 112° = [(78)^2 + (159)^2 - x^2] / (2 * 78 * 159)
=> x^2 = (78)^2 + (159)^2 - 2 * 78 * 159 * cos 112°
=> x^2 = 6084 + 25281 + 9291.74
=> x^2 = 40656.74
=> x = 201.6 miles.

Answer 2

you use the Law of Cosines: a^2 = b^2 + c^2 - 2bc cos A. "a'" is the desired distance, b is the distance the freight train has traveled, and c is the distance for bullet train has traveled.
b = 1.5 * 52 = 78; c = 1.5 * 106 = 159 and A is the angle between them = 112.
Hence, a^2 = 78^2 + 159^2 - 2(78)(159)cos 112 = 201.6

Answer 3

Train A
Speed = 52 mph
Time = 1.5 h
Distance = 78 miles

Train B
Speed = 106 mph
Time = 1.5 h
Distance = 159 miles

Apply cosine rule to find d, the distance apart.
d² = 159² + 78² - 2 (78) (159) cos 112°
d = 202 miles (to nearest whole number)

Answer 4

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