The distance between train station C and train station B is 75.02095678 miles, using the law of cosines.
In the question, we are given that a train leaves train station A and heads due west for 105 miles to reach train station B. Another train leaves train station A at the same time and travels 85 miles to train station C in a direction 45 north of west from station A.
We are asked to calculate the distance between train station C and train station B.
The given scenario can be shown by the attached diagram in the form of a triangle, where A depicts train station A, B depicts train station B, and C depicts train station C, with the angle between them being 45°.
The distance between train station A and train station B can be shown as AB = c = 105 miles.
The distance between train station A and train station C can be shown as AC = b = 85 miles.
The distance between train station B and train station C can be shown as BC = a = x miles.
By the law of cosines, in the triangle ABC, we can say that:
a² = b² + c² - 2bc cos A,
or, x² = 85² + 105² - 2(85)(105) cos 45°),
or, x² = 7225 + 11025 - 17850(0.707106781),
or, x² = 18250 - 12621.85604,
or, x² = 5628.143956,
or, x = √5628.143956 = 75.02095678.
Thus, the distance between train station C and train station B is 75.02095678 miles, using the law of cosines.
Learn more about the law of cosines at
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