Question

Two ships set sail in different directions from the same place. Tyler's ship sail 35 miles while Noah's ship sails for 42 miles. At the same time they are 56 miles apart. What was the angle between their paths when they started?

Answer 1

The angle between their paths when they started is 93°.

Explanation:

The Law of Cosines

It relates the length of the sides of a triangle with one of its internal angles.

Let a,b, and c be the length of the sides of a given triangle, and x the included angle between sides a and b, then the following relation applies:

`c^2=a^2+b^2-2ab\cos x`

When the two ships travel in different directions from the same point in the plane, they form an angle we called x in the image below.

Tyler's ship sails a=35 miles and Noah's ship sails for b=42 miles. At some time they are c=56 miles apart.

Since we know the values of all three side lengths, we solve the equation for x:

`\displaystyle \cos x=(a^2+b^2-c^2)/(2ab)`

Substituting values:

`\displaystyle \cos x=(35^2+42^2-56^2)/(2(35)(42))`

Calculating:

`\displaystyle \cos x=-(147)/(2940)=-(1)/(20)`

Computing the inverse cosine:

`x = \arccos(-0.05)`

`x \approx 93^\circ`

The angle between their paths when they started is 93°.