Question

a plane flies 40 miles from a starting point, and at the same time, a second plane flies 90 miles from the same starting point. the planes end up 67.9 miles apart. what is the angle between the two planes flight paths? 45° 110° 24° 90°

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = 40\), \(b=90\), and \(c = 67.9\). Then \(67.9^{2}=40^{2}+90^{2}-2\times40\times90\times\cos C\).

Step2: Simplify the equation

First, calculate the squares: \(4610.41=1600 + 8100-7200\cos C\). Then \(4610.41=9700-7200\cos C\).

Step3: Solve for \(\cos C\)

Rearrange the equation: \(7200\cos C=9700 - 4610.41\). So \(7200\cos C=5089.59\). Then \(\cos C=\frac{5089.59}{7200}\approx0.707\).

Step4: Find the angle \(C\)

Since \(\cos C\approx0.707\), and \(\cos45^{\circ}\approx0.707\), then \(C = 45^{\circ}\).

Answer:

\(45^{\circ}\)