### Instruction: Billy is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How far is he from his starting point? Express your answer in simplest radical form.### Response: I need to find the distance from Billy's starting point to his ending point.
I can use the law of cosines to relate the sides and angles of a triangle.
The triangle has sides of 4, 6, and x, where x is the unknown distance.
The angle opposite x is 120 degrees, since Billy turned 60 degrees northward from the east.
The law of cosines says that $x^2 = 4^2 + 6^2 - 2 \cdot 4 \cdot 6 \cdot \cos 120$.
I can simplify this by plugging in the value of $\cos 120 = -\frac{1}{2}$.
This gives me $x^2 = 16 + 36 + 24 = 76$.
Taking the square root of both sides, I get $x = \sqrt{76}$.
I can simplify the radical by factoring out the largest perfect square.
This gives me $x = \sqrt{4 \cdot 19} = 2\sqrt{19}$.