To solve this problem, we need to use trigonometry to find the height of the lighthouse, which is essentially the difference in height between the top of the lighthouse and the base of the cliff.

Steps:

1. Label the variables:

   • Let $h_1$ be the height of the cliff (from sea level to the base of the lighthouse).
   • Let $h_2$ be the height of the lighthouse (from the base of the lighthouse to its top).
   • The total height of the lighthouse from sea level to the top is $h_1 + h_2$.
   • The horizontal distance from the ship to the cliff is given as $980$ meters.

2. Using Trigonometric Ratios:

   • The angle of elevation from sea level to the base of the lighthouse is $47.3^\circ$.
   • The angle of elevation from sea level to the top of the lighthouse is $51.3^\circ$.
   • For each right triangle, we can use the tangent function since it relates the angle to the opposite side (height) and adjacent side (horizontal distance).

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

Using the tangent of $47.3^\circ$:

$\tan(47.3^\circ) = \frac{h_1}{980}$

Solve for $h_1$:

$h_1 = 980 \times \tan(47.3^\circ)$

Step 2: Calculate the total height $h_1 + h_2$ from sea level to the top of the lighthouse

Using the tangent of $51.3^\circ$:

$\tan(51.3^\circ) = \frac{h_1 + h_2}{980}$

Solve for $h_1 + h_2$:

$h_1 + h_2 = 980 \times \tan(51.3^\circ)$

Step 3: Find $h_2$ (the height of the lighthouse)

We can subtract $h_1$ from $h_1 + h_2$ to get the height of the lighthouse:

$h_2 = (h_1 + h_2) - h_1$

Let me compute this.The calculations show the following:

1. Height of the cliff ($h_1$): approximately 1062.0 meters.
2. Total height from sea level to the top of the lighthouse: approximately 1223.2 meters.
3. Height of the lighthouse ($h_2$): approximately 161.2 meters.

Thus, the height of the lighthouse from the top of the cliff is 161.2 meters, rounded to the nearest tenth.

Would you like further details or explanations on this? Here are 5 related questions:

1. How would the height change if the horizontal distance were different?
2. What is the formula for tangent, and why does it work in this case?
3. Can we calculate the angle at the top of the lighthouse using inverse trigonometric functions?
4. How can you apply this method to other real-world problems involving angles of elevation?
5. What would happen if the lighthouse were taller but the distance remained the same?

Tip: Always ensure you are working in the correct units for angles (degrees vs. radians) when using trigonometric functions in calculators or software!