To solve this problem, we'll use trigonometric relationships, particularly the tangent function, which relates the angle of elevation to the height of the lighthouse and the distances from the boat to the lighthouse.

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Step 1: Define the variables

• Height of the lighthouse beacon: $h = 111$ feet.
• From point $A$, the angle of elevation: $\theta_1 = 11^\circ$.
• From point $B$, the angle of elevation: $\theta_2 = 21^\circ$.
• Let $d_A$ and $d_B$ be the distances from points $A$ and $B$ to the lighthouse respectively.
• The distance between $A$ and $B$ (the quantity we need to find): $D = d_A - d_B$.

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Step 2: Use the tangent function for each point

For point $A$: $\tan(\theta_1) = \frac{h}{d_A}.$ Rearranging, we find: $d_A = \frac{h}{\tan(\theta_1)}.$

For point $B$: $\tan(\theta_2) = \frac{h}{d_B}.$ Rearranging, we find: $d_B = \frac{h}{\tan(\theta_2)}.$

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Step 3: Calculate the distances

Using the given data:

• $h = 111$ feet,
• $\tan(11^\circ) \approx 0.1944$,
• $\tan(21^\circ) \approx 0.3839$,

we compute: $d_A = \frac{111}{0.1944} \approx 570.8 \, \text{feet},$ $d_B = \frac{111}{0.3839} \approx 289.2 \, \text{feet}.$

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Step 4: Compute the distance between $A$ and $B$

The distance $D$ is: $D = d_A - d_B = 570.8 - 289.2 = 281.6 \, \text{feet}.$

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Final Answer:

The distance from point $A$ to point $B$ is approximately: $\boxed{281.6 \, \text{feet}}.$

Would you like further details or explanations about any step? 😊

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Related Questions:

1. How is the tangent function used to relate angles and distances in right triangles?
2. How would the result change if the height of the lighthouse were doubled?
3. What are the other trigonometric functions, and how might they apply to this problem?
4. How can this method be generalized for different elevations or distances?
5. How would the calculation differ if the lighthouse were located at a higher elevation relative to the water?

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Tip:

Always ensure that your calculator is set to the correct mode (degrees vs. radians) when working with trigonometric functions.