Question

one side of a ravine is 51 ft long, and the other is 43 ft long. a 84 ft zipline runs from the top of one side of the ravine to the other. what is the angle of the ravine? ravines angle = ?° round your answer to the nearest hundredth.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \( c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = 51\), \(b = 43\), \(c = 84\). Then \(84^{2}=51^{2}+43^{2}-2\times51\times43\times\cos C\).

Step2: Simplify the equation

First, calculate the squares: \(7056 = 2601+1849 - 4386\cos C\). Then \(7056=4450 - 4386\cos C\).

Step3: Solve for \(\cos C\)

Rearrange the equation: \(4386\cos C=4450 - 7056\). So \(4386\cos C=- 2606\). Then \(\cos C=\frac{-2606}{4386}\approx - 0.5942\).

Step4: Find the angle \(C\)

\(C=\cos^{-1}(-0.5942)\approx126.59^{\circ}\).

Answer:

\(126.59^{\circ}\)