To solve this problem, we can break it down into two steps using trigonometry. We will first determine the height of the cliff (denote it as $h_1$) and then the total height of the cliff plus the lighthouse (denote this total height as $h_2$). The difference between $h_2$ and $h_1$ will give us the height of the lighthouse from the top of the cliff.

Given:

• Distance from the ship to the base of the cliff: $d = 1100$ meters.
• Angle of elevation to the base of the lighthouse: $\theta_1 = 48.1^\circ$.
• Angle of elevation to the top of the lighthouse: $\theta_2 = 50.5^\circ$.

Step 1: Calculate the height of the cliff ($h_1$)

Using the angle of elevation to the base of the lighthouse, the height of the cliff can be calculated using the tangent function:

$\tan(\theta_1) = \frac{h_1}{d}$

Rearranging this equation to solve for $h_1$:

$h_1 = d \cdot \tan(\theta_1)$

Step 2: Calculate the total height of the cliff and lighthouse ($h_2$)

Using the angle of elevation to the top of the lighthouse, we can use a similar equation:

$\tan(\theta_2) = \frac{h_2}{d}$

Rearranging this equation to solve for $h_2$:

$h_2 = d \cdot \tan(\theta_2)$

Step 3: Calculate the height of the lighthouse

The height of the lighthouse is simply the difference between the total height ($h_2$) and the height of the cliff ($h_1$):

$\text{Height of lighthouse} = h_2 - h_1$

Let's now calculate the values using the given data.The height of the lighthouse from the top of the cliff is approximately 108.4 meters, rounded to the nearest tenth.

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

Here are some related questions:

1. How does the tangent function help in solving height problems?
2. Can this problem be solved using different trigonometric functions?
3. How would the solution change if the distance from the ship to the cliff were different?
4. What would the angles of elevation be if the lighthouse were taller or shorter?
5. How can you verify the correctness of the calculated height?

Tip: When solving trigonometric problems, always use exact values during intermediate calculations to avoid rounding errors until the final step.