Mathematics: 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?

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Two ships set sail in different directions from

lin550

10.02.2022

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

Step-by-step 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:

$\displaystyle \cos x=\frac{a^2+b^2-c^2}{2ab}$

Substituting values:

$\displaystyle \cos x=\frac{35^2+42^2-56^2}{2(35)(42)}$

Calculating:

$\displaystyle \cos x=-\frac{147}{2940}=-\frac{1}{20}$

Computing the inverse cosine:

$x = \arccos(-0.05)$

$x \approx 93^\circ$

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

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