We can solve this problem using trigonometry, specifically the tangent function, because the problem involves angles of elevation and vertical height. Here's how we'll proceed:

1. Define the variables:

   • Let the height of the lighthouse be $h = 142$ feet.
   • Let the distance from point AA to the lighthouse be $d_1$ (this is the distance we need to calculate).
   • Let the distance from point BB to the lighthouse be $d_2$ (this is the distance we need to calculate).
   • The angles of elevation from points AA and BB are $\theta_1 = 13^\circ$ and $\theta_2 = 20^\circ$, respectively.

2. Apply the tangent function: The tangent of an angle in a right triangle is the ratio of the opposite side (height of the lighthouse) to the adjacent side (distance from the boat to the lighthouse). Hence:

   $\tan(\theta_1) = \frac{h}{d_1}$ $\tan(\theta_2) = \frac{h}{d_2}$

   Substituting the known values:

   $\tan(13^\circ) = \frac{142}{d_1}$ $\tan(20^\circ) = \frac{142}{d_2}$

3. Solve for $d_1$ and $d_2$:

   $d_1 = \frac{142}{\tan(13^\circ)}$ $d_2 = \frac{142}{\tan(20^\circ)}$

   Let's calculate both distances and then find the difference between them, $d_1 - d_2$, which is the distance from point AA to point BB.

I'll calculate these values next. math

Given values

h = 142 # height of the lighthouse in feet theta1 = 13 # angle of elevation from point A in degrees theta2 = 20 # angle of elevation from point B in degrees

Convert angles to radians

theta1_rad = math.radians(theta1) theta2_rad = math.radians(theta2)

Calculate distances using the tangent function

d1 = h / math.tan(theta1_rad) d2 = h / math.tan(theta2_rad)

Find the distance between points A and B

distance_AB = d1 - d2 d1, d2, distance_ABThe distance from point AA to the lighthouse is approximately 615 feet, and the distance from point BB to the lighthouse is approximately 390 feet.

Therefore, the distance from point AA to point BB is approximately 225 feet (rounded to the nearest foot).

Would you like more details on how these calculations were made, or do you have any questions about this problem?

Here are some related questions to explore further:

1. How would the distance change if the angles of elevation were different?
2. How can you use the same method to solve for the distance in problems involving different heights?
3. What other trigonometric functions might be useful in this type of problem?
4. What would happen if the height of the lighthouse were much larger or smaller?
5. How do angles of depression relate to angles of elevation in such problems?

Tip: When using trigonometric functions like tangent in word problems, always ensure that your calculator is set to the correct mode (degrees or radians) based on the problem's angle units.