We can solve this problem by using trigonometry. The boat's path forms a triangle with the lighthouse, and the problem involves two points, $A$ and $B$, from which angles of elevation to the lighthouse are measured. Here’s how we can approach the problem:

Given:

• Horizontal distance from the boat at point $A$ to the lighthouse: 968 feet
• Angle of elevation from point $A$ to the lighthouse: 10°
• Angle of elevation from point $B$ to the lighthouse: 4°

Step-by-Step Approach:

1. Define Variables:

   • Let $h$ be the height of the lighthouse (the vertical distance to the beacon-light).
   • Let the distance from point $A$ to the lighthouse be $d_A = 968$ feet.
   • Let the distance from point $B$ to the lighthouse be $d_B$ (which we need to find).

2. Using Tangent for Point A: The tangent of the angle of elevation at point $A$ is given by: $\tan(10^\circ) = \frac{h}{968}$ Solving for $h$: $h = 968 \times \tan(10^\circ)$

3. Using Tangent for Point B: Similarly, the tangent of the angle of elevation at point $B$ is: $\tan(4^\circ) = \frac{h}{d_B}$ Solving for $d_B$: $d_B = \frac{h}{\tan(4^\circ)}$

4. Find the Distance from $A$ to $B$: The distance from point $A$ to point $B$ is simply the difference between the two distances: $\text{Distance from A to B} = d_B - 968$

Now let’s calculate these values.

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Let’s compute these values to find the answer.

math

Given values

d_A = 968 # Distance from A to lighthouse angle_A = 10 # Angle of elevation at point A angle_B = 4 # Angle of elevation at point B

Calculate the height of the lighthouse (h) using the tangent at point A

h = d_A * math.tan(math.radians(angle_A))

Calculate the distance from point B to the lighthouse using the tangent at point B

d_B = h / math.tan(math.radians(angle_B))

Find the distance from point A to point B

distance_A_to_B = d_B - d_A round(distance_A_to_B, 1) # Round to the nearest tenth of a foot The distance from point $A$ to point $B$ is approximately 1472.9 feet.

Would you like more details on the calculations, or do you have any other questions?

Here are 5 related questions:

1. How would the distance change if the angles of elevation were higher or lower?
2. What would the effect be if the boat were farther from the lighthouse initially?
3. Can this approach be applied to a more complex setup with multiple points of measurement?
4. How would you approach this problem if the angles of elevation were measured at different heights above sea level?
5. How could this method be applied to calculate distances in navigation?

Tip: Always double-check that you're using the correct units for your angles (degrees vs radians) when applying trigonometric functions.