Question

example 4: in southern california, there is a six - mile section of interstate 5 that decreases 2,500 feet in elevation as it descends grapevine hill in the tejon pass. what is the angle of depression?

Explanation:

Step1: Convert miles to feet

1 mile = 5280 feet, so 6 miles = 6×5280 = 31680 feet.

Step2: Use tangent function for angle of depression

The tangent of the angle of depression $\theta$ in a right - triangle (where the vertical change is the decrease in elevation and the horizontal change is the distance along the road) is given by $\tan\theta=\frac{\text{vertical change}}{\text{horizontal change}}$. Here, the vertical change is 2500 feet and the horizontal change is 31680 feet. So $\tan\theta=\frac{2500}{31680}\approx 0.0789$.

Step3: Find the angle

$\theta=\arctan(0.0789)$. Using a calculator, $\theta\approx 4.5^{\circ}$.

Answer:

Approximately $4.5^{\circ}$