Question

Finding the Grade of a Mountain Trail A straight trail with a uniform inclination leads from a hotel, elevation 5000 feet, to a lake in a valley, elevation 4100 feet. The length of the trail is 4100 feet. What is the inclination (grade) of the trail?

Short answer

The trail is about $12.7degree$from lake to hotel

Step-by-step solution

Step 1. Given information

Elevation of hotel$5000feet$

Elevation of valley in lake$4100feet$

Step 2. Finding the inclination of the trail 

Let $\theta$is the grade of the trail

The rise of the train is the difference between elevations

That is$5000-4100-900$

Since we know the trail is a downward slope from hotel to lake

By the defination of sine function,

Know the formula

$$\begin{gathered} \sin \theta =\frac{opp}{hyp} \\ \sin \theta =\frac{900}{4100} \\ \theta ={\sin }^{-1}\left(\frac{900}{4100}\right) \\ \theta =12.7 \end{gathered}$$

The trail is about$12.7degree$from lake to hotel