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This problem can be solved using trigonometry, particularly by applying the Law of Sines and basic trigonometric relationships. Let's break it down step by step.

### Given:
- The distance \( AB = 18.4 \) m.
- The angle at \( A = 58^\circ \).
- The angle at \( B = 72^\circ \).
- \( D \) represents the height of the cliff.

We are asked to find the height \( CD \) of the cliff, to the nearest tenth of a meter.

### Step 1: Find angle \( C \)
Since the sum of the angles in a triangle is always \( 180^\circ \), we can find \( \angle C \).

\[
\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 58^\circ - 72^\circ = 50^\circ
\]

### Step 2: Apply the Law of Sines
We can use the Law of Sines to find the distance \( AC \), which will be useful for determining the height.

\[
\frac{AC}{\sin B} = \frac{AB}{\sin C}
\]

Substitute the known values:

\[
\frac{AC}{\sin 72^\circ} = \frac{18.4}{\sin 50^\circ}
\]

Solve for \( AC \):

\[
AC = \frac{18.4 \times \sin 72^\circ}{\sin 50^\circ}
\]

Using a calculator:

\[
AC \approx \frac{18.4 \times 0.9511}{0.7660} \approx 22.84 \, \text{m}
\]

### Step 3: Find the height \( CD \)
Now that we know \( AC \), we can use trigonometry to find \( CD \), which is the height of the cliff.

In triangle \( ACD \), the height \( CD \) can be found using:

\[
CD = AC \times \sin 58^\circ
\]

Substitute the value of \( AC \):

\[
CD = 22.84 \times \sin 58^\circ
\]

\[
CD \approx 22.84 \times 0.848 \approx 19.4 \, \text{m}
\]

### Final Answer:
The height of the cliff is approximately **19.4 meters**, rounded to the nearest tenth.

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Do you want more details or have any questions? Here are five related questions you could explore next:

1. How does the Law of Cosines compare to the Law of Sines in solving triangle problems?
2. How would the solution change if the distance across the river were doubled?
3. What other methods, besides trigonometry, could be used to find the height of the cliff?
4. How does the accuracy of angles affect the final height calculation?
5. Could this method be used to measure the height of a building? Why or why not?

**Tip:** Always check if the sum of the angles in a triangle adds up to 180° to avoid calculation mistakes.

The diagram shows the readings that were taken in order to determine the height of a cliff on the other side of a river. What is the height of the cliff, to the nearest tenth of a meter?

Learn how to determine the height of a cliff using trigonometry and the Law of Sines. This detailed solution includes steps on calculating angles and applying trigonometric relationships to find the height of the cliff across a river. Suitable for high school students in Grades 10-12.

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