Question

solving a word problem using
o airplanes leave an airport at the same time. an hour later, the planes are 233 km apart. if one plane has traveled 208 km and the other has traveled 231 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 \(c^{2}=a^{2}+b^{2}-2ab\cos C\). In this triangle, let \(a = 208\), \(b=231\), \(c = 233\), and \(C=\theta\). Then \(233^{2}=208^{2}+231^{2}-2\times208\times231\times\cos\theta\).
First, calculate the squares: \(233^{2}=54289\), \(208^{2}=43264\), \(231^{2}=53361\).
Substitute into the formula: \(54289=43264 + 53361-2\times208\times231\times\cos\theta\).
Simplify the right - hand side: \(54289=96625-96144\cos\theta\).

Step2: Solve for \(\cos\theta\)

Rearrange the equation: \(96144\cos\theta=96625 - 54289\).
\(96144\cos\theta=42336\).
Then \(\cos\theta=\frac{42336}{96144}\approx0.4403\).

Step3: Find \(\theta\)

Take the inverse cosine: \(\theta=\cos^{-1}(0.4403)\).
Using a calculator, \(\theta\approx63.9^{\circ}\).

Answer:

\(63.9^{\circ}\)