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The question involves trigonometry. Let me summarize and solve the problem:

### Problem Setup:
- The **height of the cliff** is \( 80 \, \text{m} \) above the lakeshore.
- The surveyor observes a boat on the lake at an **angle of depression** of \( 38^\circ \).
- We are to find the **horizontal distance** (base of the right triangle) between the boat and the base of the cliff.

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### Solution:
We can use the **tangent** function in trigonometry, which relates the angle of depression \( \theta \), the height of the cliff \( h \), and the horizontal distance \( x \):

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Here:
- \( \theta = 38^\circ \) (angle of depression),
- \( \text{opposite} = 80 \, \text{m} \) (height of the cliff),
- \( \text{adjacent} = x \) (horizontal distance we need to find).

Rearranging for \( x \):

\[
x = \frac{\text{opposite}}{\tan(\theta)} = \frac{80}{\tan(38^\circ)}.
\]

Using a calculator to find \( \tan(38^\circ) \):

\[
\tan(38^\circ) \approx 0.7813.
\]

Now substitute back:

\[
x = \frac{80}{0.7813} \approx 102.4 \, \text{m}.
\]

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### Final Answer:
The horizontal distance from the boat to the base of the cliff is approximately **102.4 meters**.

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Would you like me to provide a step-by-step diagram explanation? Let me know!

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Here are 5 related questions you might find useful:
1. What is the horizontal distance if the angle of depression is increased to \( 45^\circ \)?
2. How would the result change if the cliff height were doubled?
3. Can you calculate the angle of depression if the distance to the boat is \( 120 \, \text{m} \)?
4. What trigonometric function would you use if the surveyor measured the hypotenuse distance instead?
5. How would the scenario differ if the surveyor were at a lower height?

**Tip:** Always double-check which trigonometric function to use—**tangent** relates height and horizontal distance, while sine and cosine involve hypotenuse measurements.

The highest point along a cliff is 80 m above the lakeshore. A surveyor stands on the top of the cliff, looking through a 1.5 m tall transit instrument. He spots a boat out on the lake at an angle of depression of 38°. How far, to the nearest tenth of a meter, is it from the boat to the base of the cliff?

Solve a real-world trigonometry problem involving an 80-meter-high cliff and an angle of depression of 38°. Learn how to calculate the horizontal distance to the base of the cliff using the tangent function. Perfect for students in Grades 9-11.

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