A lighthouse sits at the edge of a cliff, as shown. A ship at sea level is 980 meters from the base of the cliff. The angle of elevation from sea level to the base of the lighthouse is 47.3\deg . The angle of elevation from sea level to the top of the lighthouse is 51.3\deg . Find the height of the lighthouse from the top of the cliff.
Do not round any intermediate computations. Round your answer to the nearest tenth.
Note that the figure below is not drawn to scale.

To solve this problem, we'll break it down into steps using trigonometry. Here's a step-by-step approach:

### Step 1: Define the variables
- Let \( h_1 \) be the height of the cliff.
- Let \( h_2 \) be the height of the lighthouse from the top of the cliff.
- The total height of the lighthouse from sea level is \( h_1 + h_2 \).

### Step 2: Use the angle of elevation to the base of the lighthouse
Using the tangent function for the angle of elevation to the base of the lighthouse:
\[
\tan(47.3^\circ) = \frac{h_1}{980}
\]
\[
h_1 = 980 \times \tan(47.3^\circ)
\]

### Step 3: Use the angle of elevation to the top of the lighthouse
Similarly, using the tangent function for the angle of elevation to the top of the lighthouse:
\[
\tan(51.3^\circ) = \frac{h_1 + h_2}{980}
\]
\[
h_1 + h_2 = 980 \times \tan(51.3^\circ)
\]

### Step 4: Solve for \( h_2 \)
We can find \( h_2 \) by subtracting the equation for \( h_1 \) from the equation for \( h_1 + h_2 \):
\[
h_2 = 980 \times \tan(51.3^\circ) - 980 \times \tan(47.3^\circ)
\]

### Step 5: Calculate the value of \( h_2 \)
Let's compute the values.

\[
h_1 = 980 \times \tan(47.3^\circ)
\]
\[
h_2 = 980 \times (\tan(51.3^\circ) - \tan(47.3^\circ))
\]

Now, I'll perform the calculations.The height of the lighthouse from the top of the cliff is approximately **161.2 meters**, rounded to the nearest tenth.

Would you like more details or have any questions? 

Here are some related questions:
1. How would the result change if the distance from the ship to the cliff increased?
2. What if the angles of elevation were smaller? How would that affect the height?
3. Can you determine the total height of the lighthouse from sea level?
4. How can this method be applied to find the height of a mountain?
5. What would happen if the angle to the base and the top were the same?

**Tip:** When solving trigonometric problems involving angles of elevation, always ensure your calculator is in the correct mode (degrees or radians).

A lighthouse sits at the edge of a cliff, as shown. A ship at sea level is 980 meters from the base of the cliff. The angle of elevation from sea level to the base of the lighthouse is 47.3°. The angle of elevation from sea level to the top of the lighthouse is 51.3°. Find the height of the lighthouse from the top of the cliff. Do not round any intermediate computations. Round your answer to the nearest tenth.

This trigonometry problem involves using angles of elevation to find the height of a lighthouse from the top of a cliff. The ship is 980 meters away, and the angles of elevation to the base and top of the lighthouse are 47.3° and 51.3°, respectively. Learn how to apply the tangent function to solve this problem with step-by-step guidance, suitable for Grades 10-12.

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