A boat is heading towards a lighthouse, whose beacon-light is 127 feet above the water. From point A*A*, 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 B*B* at some later time to be 24degrees∘. Find the distance from point A*A* to point B*B*. Round your answer to the nearest tenth of a foot if necessary.

To solve for the distance between points \(A\) and \(B\), we will use trigonometry, specifically the tangent function for the angles of elevation from points \(A\) and \(B\).

Let:
- \(h = 127\) feet (the height of the lighthouse beacon above the water)
- \(\theta_A = 12^\circ\) (angle of elevation at point \(A\))
- \(\theta_B = 24^\circ\) (angle of elevation at point \(B\))
- \(d_A\) = distance from point \(A\) to the base of the lighthouse
- \(d_B\) = distance from point \(B\) to the base of the lighthouse
- \(d_{AB}\) = distance between points \(A\) and \(B\) (the distance we need to find)

### Step 1: Calculate the distance \(d_A\)

Using the tangent function at point \(A\):
\[
\tan(12^\circ) = \frac{h}{d_A}
\]
Solving for \(d_A\):
\[
d_A = \frac{h}{\tan(12^\circ)}
\]
Using a calculator:
\[
\tan(12^\circ) \approx 0.2126
\]
\[
d_A = \frac{127}{0.2126} \approx 597.4 \text{ feet}
\]

### Step 2: Calculate the distance \(d_B\)

Using the tangent function at point \(B\):
\[
\tan(24^\circ) = \frac{h}{d_B}
\]
Solving for \(d_B\):
\[
d_B = \frac{127}{\tan(24^\circ)}
\]
Using a calculator:
\[
\tan(24^\circ) \approx 0.4452
\]
\[
d_B = \frac{127}{0.4452} \approx 285.3 \text{ feet}
\]

### Step 3: Calculate the distance \(d_{AB}\)

The distance between points \(A\) and \(B\) is the difference between \(d_A\) and \(d_B\):
\[
d_{AB} = d_A - d_B
\]
\[
d_{AB} = 597.4 - 285.3 \approx 312.1 \text{ feet}
\]

Thus, the distance from point \(A\) to point \(B\) is approximately **312.1 feet**.

Would you like more details or have any questions?

Here are 8 related questions:
1. How do you find the height of an object using the tangent function?
2. What is the tangent function and how is it used in angle elevation problems?
3. How do you determine distances using angles of elevation and trigonometry?
4. What happens to the angles of elevation as you get closer to the lighthouse?
5. Can you explain the significance of using two different angles of elevation?
6. How can this method be applied to find the distance between two points in other scenarios?
7. What are the limitations of using trigonometry for these types of problems?
8. How can you ensure the accuracy of your trigonometric calculations?

**Tip:** Always double-check the unit mode on your calculator (degrees or radians) when performing trigonometric calculations to ensure accurate results.

Learn how to calculate the distance between two points A and B using trigonometry and the tangent function. This problem involves measuring angles of elevation to determine distances, suitable for advanced high school students.

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