Let's break this problem down using trigonometry. The scenario involves two points, $A$ and $B$, where Samantha measures the angles of elevation to an airplane at two different times. We are asked to find the distance the plane traveled between these points, given the altitude of the plane remains constant at 6375 feet.

Step 1: Visualizing the situation

We have two right-angled triangles:

• At point $A$, the angle of elevation is $15^\circ$.
• At point $B$, the angle of elevation is $28^\circ$.
• The altitude of the plane is $6375$ feet (constant for both points).

The key is to use the tangent function for each of the angles, which relates the angle of elevation to the vertical height and horizontal distance. Specifically: $\tan(\theta) = \frac{\text{height}}{\text{horizontal distance}}$ Let the horizontal distances from points $A$ and $B$ to the point directly beneath the plane (i.e., where the plane will pass overhead) be $x_A$ and $x_B$, respectively.

Step 2: Setting up the equations

At point $A$: $\tan(15^\circ) = \frac{6375}{x_A}$ So, $x_A = \frac{6375}{\tan(15^\circ)}$

At point $B$: $\tan(28^\circ) = \frac{6375}{x_B}$ So, $x_B = \frac{6375}{\tan(28^\circ)}$

Step 3: Finding the distance the plane traveled

The plane traveled the horizontal distance between points $A$ and $B$. So, we need to calculate $x_A - x_B$, the difference between the two horizontal distances.

Step 4: Calculating the values

First, we need to compute the tangent values and plug them into the equations:

• $\tan(15^\circ) \approx 0.2679$
• $\tan(28^\circ) \approx 0.5317$

Now, calculate the distances: $x_A = \frac{6375}{0.2679} \approx 23883.5 \, \text{feet}$ $x_B = \frac{6375}{0.5317} \approx 11968.4 \, \text{feet}$

Step 5: Finding the traveled distance

The distance the plane traveled from point $A$ to point $B$ is: $\text{Distance} = x_A - x_B = 23883.5 - 11968.4 = 11915.1 \, \text{feet}$

Final Answer:

The plane traveled approximately 11,915.1 feet from point $A$ to point $B$.

Would you like any further explanation or clarification on this? Here are some related questions you might find interesting:

1. How would the distance change if the altitude of the plane were different?
2. How does the angle of elevation relate to the speed of the airplane?
3. What would happen if the plane's path was not straight?
4. How could we solve this problem using vectors instead of trigonometry?
5. How does the tangent function change with different angles of elevation?

Tip: When working with problems involving angles and distances, always make sure your calculator is in the correct mode (degrees or radians) based on the problem.