Question

one side of a ravine is 62 ft long, and the other is 50 ft long. a 103 ft zipline runs from the top of one side of the ravine to the other. what is the angle of the ravine?
ravine’s angle = ?°
round your answer to the nearest hundredth.

Explanation:

Step1: Identify the Law to Use

We have a triangle with two sides (62 ft, 50 ft) and the included angle opposite the zipline (103 ft). So, use the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( c = 103 \), \( a = 62 \), \( b = 50 \), and \( C \) is the angle of the ravine.

Step2: Rearrange the Formula for \( \cos(C) \)

From \( 103^2 = 62^2 + 50^2 - 2(62)(50)\cos(C) \), we get:
\( \cos(C) = \frac{62^2 + 50^2 - 103^2}{2(62)(50)} \)

Step3: Calculate the Numerator and Denominator

Numerator: \( 62^2 + 50^2 - 103^2 = 3844 + 2500 - 10609 = -4265 \)
Denominator: \( 2(62)(50) = 6200 \)
So, \( \cos(C) = \frac{-4265}{6200} \approx -0.6879 \)

Step4: Find the Angle \( C \)

\( C = \arccos(-0.6879) \approx 133.47^\circ \) (using a calculator in degree mode)

Answer:

\( 133.47 \)