Let's solve both parts of this problem:

EX 2

The angle of depression from the top of a ski hill to the bottom is given as $24.6^\circ$, and the length of the hill (hypotenuse) is $168 \, \text{m}$.

To find the drop in altitude (vertical distance), we can use trigonometry:

1. Let the vertical drop be $h$.
2. Using the sine function, since $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$: $\sin(24.6^\circ) = \frac{h}{168}$
3. Solving for $h$: $h = 168 \times \sin(24.6^\circ)$
4. Calculating $h$: $h \approx 168 \times 0.4162 \approx 69.9 \, \text{m}$

So, the drop in altitude of the ski hill is approximately 69.9 meters.

You Try

An airplane is flying at an altitude of $400 \, \text{m}$ and is a horizontal distance of $2300 \, \text{m}$ from an airstrip. We need to find the angle of depression from the airplane to the airstrip.

1. Let the angle of depression be $\theta$.
2. Using the tangent function, since $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$: $\tan(\theta) = \frac{400}{2300}$
3. Solving for $\theta$: $\theta = \tan^{-1} \left( \frac{400}{2300} \right)$
4. Calculating $\theta$: $\theta \approx \tan^{-1}(0.1739) \approx 9.9^\circ$

So, the angle of depression from the airplane to the airstrip is approximately 9.9 degrees.

Would you like more details on these calculations or have any questions?

Here are five related questions to expand your understanding:

1. How would the drop in altitude change if the angle of depression of the ski hill was increased?
2. If the horizontal distance to the bottom of the ski hill is given, how could you use it to find the vertical drop?
3. What other trigonometric functions can be used in right triangles and how would they apply here?
4. How does the tangent function relate to real-life applications, like angles of elevation and depression?
5. What would the calculations look like if the distance between the airplane and the airstrip was reduced?

Tip: For angle of depression or elevation problems, identifying the correct trigonometric function based on known sides (opposite, adjacent, or hypotenuse) can make solving easier!