# Two trains leave the same train station at the same time, moving along straight tracks that form 35 angle. If one train travels at an average speed of 100 mi/hr and the other at an average speed of 90 mi/hr, how far apart are the trains after 30 min?

Nov 3, 2016

Trains are $28.96$ miles apart after $30$ minutes.

#### Explanation:

The train with $100$ mi/hr has moved in $30$ min $\frac{100}{\frac{30}{60}} = 50$miles
The train with $90$ mi/hr has moved in $30$ min $\frac{90}{\frac{30}{60}} = 45$miles.
So this may be regarded as a triangle of which sides are a=50 ; b=45 and their included angle is $\angle C = {35}^{0}$

by applying cosine law, ${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos C \mathmr{and} {c}^{2} = {50}^{2} + {45}^{2} - 2 \cdot 50 \cdot 45 \cdot \cos 35 = 2500 + 2025 + 4500 \cdot 0.82 = 838.82 \left(2 \mathrm{dp}\right)$
$c = \sqrt{838.82} = 28.96$ miles.

The trains are $28.96$ miles apart after $30$ minutes. [Ans]