A plane is flying within sight of the Gateway Arch in St. Louis, Missouri, at an elevation of 40,000 ft. The pilot would like to estimate her distance from the Gateway Arch. She finds that the angle of depression to a point on the ground below the arch is 19°. (Round your answers to the nearest foot.)
(a) What is the distance between the plane and the base of the arch?

(b) What is the distance between a point on the ground directly below the plane and the base of the arch?

To solve this problem, we can use trigonometric principles, specifically involving the angle of depression and the distances involved.

Let's break it down into two parts:

### Part (a): Distance between the plane and the base of the arch

1. **Define the variables:**
   - Let \( h \) be the height of the plane above the ground, which is 40,000 ft.
   - Let \( \theta \) be the angle of depression from the plane to the base of the arch, which is 19°.
   - Let \( d \) be the distance between the plane and the base of the arch.

2. **Use the trigonometric relationship:**
   - The angle of depression is the same as the angle of elevation from the ground to the plane.
   - The relationship involves the hypotenuse \( d \) and the adjacent side \( h \) of the right triangle formed.

   \[
   \cos(\theta) = \frac{h}{d}
   \]

3. **Solve for \( d \):**

   \[
   d = \frac{h}{\cos(\theta)}
   \]

4. **Substitute the known values:**

   \[
   d = \frac{40000}{\cos(19^\circ)}
   \]

5. **Calculate the value:**

   \[
   \cos(19^\circ) \approx 0.9455
   \]

   \[
   d = \frac{40000}{0.9455} \approx 42299 \text{ ft}
   \]

### Part (b): Distance between the point on the ground directly below the plane and the base of the arch

1. **Define the variables:**
   - Let \( x \) be the distance between the point on the ground directly below the plane and the base of the arch.

2. **Use the trigonometric relationship involving the tangent function:**

   \[
   \tan(\theta) = \frac{h}{x}
   \]

3. **Solve for \( x \):**

   \[
   x = \frac{h}{\tan(\theta)}
   \]

4. **Substitute the known values:**

   \[
   x = \frac{40000}{\tan(19^\circ)}
   \]

5. **Calculate the value:**

   \[
   \tan(19^\circ) \approx 0.3443
   \]

   \[
   x = \frac{40000}{0.3443} \approx 116205 \text{ ft}
   \]

### Summary

- **(a)** The distance between the plane and the base of the arch is approximately 42,299 ft.
- **(b)** The distance between the point on the ground directly below the plane and the base of the arch is approximately 116,205 ft.

Would you like more details or have any questions? Here are five more questions you might be interested in:

1. How would the distances change if the plane's elevation were increased to 50,000 ft?
2. Can we use a different trigonometric function to solve the problem?
3. How does the curvature of the Earth affect these calculations?
4. What if the angle of depression were measured incorrectly? How would that affect the distances?
5. How can we verify these calculations using different methods?

**Tip:** When dealing with trigonometric problems, always double-check the angle values and their respective functions (sine, cosine, tangent) to ensure accuracy in your calculations.

Explore how to calculate the distance between a plane at 40,000 ft and the Gateway Arch in St. Louis using trigonometric functions. This problem includes an angle of depression of 19°, leading to distances of approximately 42,299 ft and 116,205 ft. Ideal for high school students.

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