To find the distance the plane traveled from point A to point B, we can use trigonometry. Let's define the given information:

• The plane's constant altitude: $h = 7125$ feet.
• The angle of elevation at point A: $\theta_A = 18^\circ$.
• The angle of elevation at point B: $\theta_B = 31^\circ$.

Step 1: Define the Distance Relationships

Let:

• $x_A$ be the horizontal distance from Nevaeh at point A to the point directly below the plane.
• $x_B$ be the horizontal distance from Nevaeh at point B to the point directly below the plane.
• $d$ be the distance the plane traveled from point A to point B, which is the difference $d = x_A - x_B$.

From right triangle trigonometry: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Thus, for points A and B: $x_A = \frac{h}{\tan(\theta_A)} = \frac{7125}{\tan(18^\circ)}$ $x_B = \frac{h}{\tan(\theta_B)} = \frac{7125}{\tan(31^\circ)}$

Step 2: Compute the Distances

Now, calculating these values:

$x_A = \frac{7125}{\tan(18^\circ)}$ $x_B = \frac{7125}{\tan(31^\circ)}$

Let’s compute these: math

Given values

h = 7125 # Altitude of the plane in feet theta_A = math.radians(18) # Convert degrees to radians theta_B = math.radians(31) # Convert degrees to radians

Calculate horizontal distances

x_A = h / math.tan(theta_A) x_B = h / math.tan(theta_B)

Distance traveled by the plane

d = x_A - x_B dThe distance the plane traveled from point A to point B is approximately 10,070.5 feet.

Would you like a step-by-step breakdown of the calculations?

Here are five related questions to explore further:

1. How would the distance change if the altitude of the plane increased?
2. What would happen if the angles of elevation were measured from different heights?
3. How can this problem be solved using the Law of Sines instead?
4. What is the significance of using tangent in this type of problem?
5. How would the solution change if Nevaeh were moving instead of stationary?

Tip:

In trigonometry problems involving right triangles, always check whether tangent, sine, or cosine is the best function to use based on the given information.