To solve this problem, we can use trigonometry to determine the distance between points $A$ and $B$. Here’s a step-by-step approach:

Given:

• Height of the lighthouse ($h$) = 104 feet.
• Angle of elevation from point $A$ ($\theta_1$) = 11°.
• Angle of elevation from point $B$ ($\theta_2$) = 22°.

To Find:

• The distance between points $A$ and $B$.

Approach:

1. Define the distances:

   • Let $x$ be the distance from point $A$ to the base of the lighthouse.
   • Let $y$ be the distance from point $B$ to the base of the lighthouse.
   • The distance between $A$ and $B$ is $x - y$.

2. Use trigonometric relationships: From point $A$, using the tangent of angle $\theta_1$: $\tan(11^\circ) = \frac{h}{x}$ So, $x = \frac{104}{\tan(11^\circ)}$.

   From point $B$, using the tangent of angle $\theta_2$: $\tan(22^\circ) = \frac{h}{y}$ So, $y = \frac{104}{\tan(22^\circ)}$.

3. Find the distance between $A$ and $B$: $\text{Distance between } A \text{ and } B = x - y = \frac{104}{\tan(11^\circ)} - \frac{104}{\tan(22^\circ)}$

Calculation:

Now, let's calculate this using a calculator for the tangent values:

• $\tan(11^\circ) \approx 0.1944$
• $\tan(22^\circ) \approx 0.4040$

$x = \frac{104}{0.1944} \approx 535.12 \text{ feet}$ $y = \frac{104}{0.4040} \approx 257.43 \text{ feet}$

$\text{Distance between } A \text{ and } B = 535.12 - 257.43 \approx 278 \text{ feet}$

Final Answer:

The distance from point $A$ to point $B$ is approximately 278 feet.

Would you like more details or have any questions about this solution?

Related Questions:

1. How would the answer change if the height of the lighthouse was 200 feet instead?
2. What is the significance of the angle of elevation in trigonometric problems?
3. How can we verify this solution using a different trigonometric function?
4. What is the formula for tangent and how is it used in right triangle problems?
5. If the boat continued to move, what would the angle of elevation be at a distance of 100 feet from point $B$?

Tip:

In problems involving right triangles and angles of elevation, using tangent is often helpful since it relates the height of an object to the distance from the object.