Answer:
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:
[tex]c^2=a^2+b^2-2ab\cos x[/tex]
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:
[tex]\displaystyle \cos x=\frac{a^2+b^2-c^2}{2ab}[/tex]
Substituting values:
[tex]\displaystyle \cos x=\frac{35^2+42^2-56^2}{2(35)(42)}[/tex]
Calculating:
[tex]\displaystyle \cos x=-\frac{147}{2940}=-\frac{1}{20}[/tex]
Computing the inverse cosine:
[tex]x = \arccos(-0.05)[/tex]
[tex]x \approx 93^\circ[/tex]
The angle between their paths when they started is 93°.
