Question

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. To the nearest mile, how far is train station C from train station B?
75 miles
109 miles
135 miles
176 miles

Answer 1

Answer:

Option (a) is correct.

To the nearest mile, the distance between train station C from train station B is 75 miles.

Step-by-step explanation:

 Given : 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 have to find the distance between train station C from train station B.

We first draw the position of stations according to given statement.

Figure shown below in attachment.

Since we have to find the distance BC , Let BC = x

For a triangle ABC, with ∠A = 45° and AB = 105 m and AC = 85 m

Using Cosine rule,

$BC=\sqrt{CA^2+AB^2-2\cdot AB\cdot AC\cdot \cos A }$

Substitute, we have,

$x=\sqrt{(85)^2+(105)^2-2\cdot 85\cdot 105\cdot \cos 45^{\circ}}$

Simplify , we have,

$x=75.020$

Thus, to the nearest mile, the distance between train station C from train station B is 75 miles.

Answer 2

Hello,

Answer A

Theorem of Al Kashi gives:

x²=105²+85²-2*105*85*√2/2=5628.14395...
x=75.0209... (mi)