Question

three boats are out at sea: boat a, boat b, and boat c. the crew of boat a can see both boat b and boat c. the angle between the line of sight to boat b and the line of sight to boat c is 49°. if the distance between boat a and boat b is 1.8 miles and the distance between boat a and boat c is 3.2 miles, what is the distance between boat b and boat c?

distance = 2.43 mi

Explanation:

Step1: Identify the formula

We use the Law of Cosines. For a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). Let the distance between Boat A and Boat B be \(a = 1.8\) miles, the distance between Boat A and Boat C be \(b=3.2\) miles, and the included - angle \(\theta = 49^{\circ}\), and the distance between Boat B and Boat C be \(c\).

Step2: Substitute values into the formula

$$
LATEXBLOCK0
$$

We know that \(\cos(49^{\circ})\approx0.656\).

$$
LATEXBLOCK1
$$

Step3: Solve for \(c\)

Take the square - root of both sides: \(c=\sqrt{5.913}\approx2.43\) miles.

Answer:

\(2.43\) miles