Question

two airplanes leave an airport at the same time. an hour later, the planes are 218 km apart. if one plane has traveled 248 km and the other has traveled 280 km during the hour, find the angle θ between their flight paths. see the figure below. carry your intermediate computations to at least four decimal places. round your answer to the nearest tenth of a degree.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). Let \(a = 248\), \(b=280\), \(c = 218\). Then \(218^{2}=248^{2}+280^{2}-2\times248\times280\times\cos\theta\).

Step2: Simplify the equation

First, calculate the squares: \(218^{2}=47524\), \(248^{2}=61504\), \(280^{2}=78400\).
The equation becomes \(47524=61504 + 78400-138880\cos\theta\).
Combine the right - hand side: \(47524=139904-138880\cos\theta\).

Step3: Solve for \(\cos\theta\)

Rearrange the equation: \(138880\cos\theta=139904 - 47524\).
\(138880\cos\theta=92380\).
\(\cos\theta=\frac{92380}{138880}\approx0.6651\).

Step4: Find \(\theta\)

\(\theta=\cos^{-1}(0.6651)\approx48.3^{\circ}\).

Answer:

\(48.3^{\circ}\)