A boat is heading towards a lighthouse, whose beacon-light is 150 feet above the water. From point AA, the boat’s crew measures the angle of elevation to the beacon, 12degrees 
∘
 , before they draw closer. They measure the angle of elevation a second time from point BB at some later time to be 21degrees 
∘
 . Find the distance from point AA to point BB. Round your answer to the nearest foot if necessary.

To solve this problem, we can use trigonometry, specifically the tangent function, since we have angles of elevation and the height of the lighthouse.

### Step 1: Define the problem

Let:
- \( h = 150 \) feet (the height of the lighthouse).
- \( \theta_1 = 12^\circ \) (the angle of elevation from point AA).
- \( \theta_2 = 21^\circ \) (the angle of elevation from point BB).
- \( d_1 \) be the distance from point AA to the base of the lighthouse.
- \( d_2 \) be the distance from point BB to the base of the lighthouse.
- \( d \) be the distance from point AA to point BB.

### Step 2: Set up the equations

From point AA:
\[
\tan(12^\circ) = \frac{h}{d_1} \quad \Rightarrow \quad d_1 = \frac{h}{\tan(12^\circ)}
\]

From point BB:
\[
\tan(21^\circ) = \frac{h}{d_2} \quad \Rightarrow \quad d_2 = \frac{h}{\tan(21^\circ)}
\]

### Step 3: Calculate the distances

\[
d_1 = \frac{150}{\tan(12^\circ)}
\]
\[
d_2 = \frac{150}{\tan(21^\circ)}
\]

### Step 4: Find the distance from point AA to point BB

The distance between AA and BB is the difference between \( d_1 \) and \( d_2 \):
\[
d = d_1 - d_2
\]

Now, let's calculate these values.

First, calculate \( d_1 \):
\[
d_1 = \frac{150}{\tan(12^\circ)} \approx \frac{150}{0.2126} \approx 705.82 \text{ feet}
\]

Next, calculate \( d_2 \):
\[
d_2 = \frac{150}{\tan(21^\circ)} \approx \frac{150}{0.3839} \approx 390.72 \text{ feet}
\]

Finally, calculate \( d \):
\[
d = d_1 - d_2 \approx 705.82 - 390.72 \approx 315.10 \text{ feet}
\]

### Final Answer
The distance from point AA to point BB is approximately **315 feet**.

Would you like more details or have any questions? 

Here are 5 related questions:
1. How do different angles of elevation affect the distance calculation?
2. What other trigonometric functions could be used in similar problems?
3. How would the problem change if the lighthouse height was unknown?
4. How would the solution be affected if the angles were measured from the top of a hill instead of water level?
5. Can the method used here apply to other scenarios, such as aircraft approaching a tower?

**Tip:** In trigonometry, always ensure your calculator is set to the correct mode (degrees/radians) based on the angle measurements provided.

Learn how to calculate the distance between two points using trigonometry with an angle of elevation problem. Solve for the distance from point AA to BB given angles of elevation, lighthouse height, and trigonometric functions. Suitable for high school students.

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