We can solve this problem using trigonometry. The situation forms two right triangles, where the plane's altitude is constant at 58,755,875 feet, and Kayden's observations at two points form angles of elevation of 16° and 33°, respectively.

Let's break this down:

Step 1: Setup the problem using trigonometric relationships

• The height of the plane is constant, so we can think of this as the vertical side of a right triangle.
• The horizontal distances from Kayden's two observation points (Point A and Point B) to the point on the ground directly beneath the plane form the bases of the triangles.

Let:

• $h = 58,755,875$ feet (altitude of the plane),
• $\theta_A = 16^\circ$ (angle of elevation at Point A),
• $\theta_B = 33^\circ$ (angle of elevation at Point B),
• $d_A$ = distance from Point A to the point directly beneath the plane,
• $d_B$ = distance from Point B to the point directly beneath the plane.

Step 2: Use the tangent function

In both right triangles, the tangent of the angle of elevation relates the height $h$ to the horizontal distance.

For Point A: $\tan(\theta_A) = \frac{h}{d_A} \quad \Rightarrow \quad d_A = \frac{h}{\tan(\theta_A)}$

For Point B: $\tan(\theta_B) = \frac{h}{d_B} \quad \Rightarrow \quad d_B = \frac{h}{\tan(\theta_B)}$

Step 3: Calculate $d_A$ and $d_B$

We will now compute both distances:

$d_A = \frac{58,755,875}{\tan(16^\circ)}$ $d_B = \frac{58,755,875}{\tan(33^\circ)}$

Step 4: Find the distance traveled by the plane

The plane travels the horizontal distance from Point A to Point B, which is the difference in $d_A$ and $d_B$: $\text{Distance traveled} = d_A - d_B$

Let's now calculate these values.

math

Given values

h = 58755875 # altitude of the plane in feet theta_A_deg = 16 # angle of elevation at point A in degrees theta_B_deg = 33 # angle of elevation at point B in degrees

Convert angles to radians

theta_A_rad = math.radians(theta_A_deg) theta_B_rad = math.radians(theta_B_deg)

Calculate the horizontal distances using the tangent function

d_A = h / math.tan(theta_A_rad) d_B = h / math.tan(theta_B_rad)

Calculate the distance the plane traveled

distance_traveled = d_A - d_B round(distance_traveled, 1)The distance the plane traveled from point A to point B is approximately 114,429,973.8 feet.

Would you like more details on how this solution was reached, or do you have any further questions?

Here are some related questions to expand your understanding:

1. How does changing the altitude of the plane affect the distance traveled?
2. What would happen if the angle of elevation at Point A were 45° instead of 16°?
3. How can you apply this method to other real-life scenarios like surveying or navigation?
4. What is the relationship between the angles of elevation and the horizontal distances in this type of problem?
5. How do trigonometric functions like sine, cosine, and tangent relate to real-world distances?

Tip: When working with trigonometric functions, remember to always convert degrees to radians if you're using a calculator set to radian mode!