From the top of a vertical cliff 125 m high, the angle of depression of a boat is 18.833o. Find the distance of the boat from the cliff.

 	
Select an answer:
 	
360.5

 	
266.5

 	
376.5

 	
366.5

We are given the following information:

- Height of the cliff = 125 m
- Angle of depression = \(18.833^\circ\)
- We need to find the horizontal distance of the boat from the base of the cliff.

### Step 1: Identify the right triangle components
The angle of depression is measured from the horizontal line at the top of the cliff. This creates a right-angled triangle where:
- The height of the cliff is the opposite side (\(O = 125 \, \text{m}\)).
- The horizontal distance from the boat to the base of the cliff is the adjacent side (\(A\)).
- The angle of depression (\(\theta = 18.833^\circ\)) is the angle between the horizontal and the line of sight to the boat.

### Step 2: Use the tangent function
The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
\[
\tan(\theta) = \frac{O}{A}
\]
Substitute the given values:
\[
\tan(18.833^\circ) = \frac{125}{A}
\]
Solving for \(A\):
\[
A = \frac{125}{\tan(18.833^\circ)}
\]

### Step 3: Calculate the distance
Let me calculate this value.The horizontal distance of the boat from the cliff is approximately \(366.5\) meters.

So, the correct answer is:

**366.5**

Do you want further details or have any questions?

Here are five related questions to explore:
1. How would the horizontal distance change if the cliff height increased?
2. What if the angle of depression were smaller, say \(10^\circ\)? How would that affect the distance?
3. Can you calculate the distance of the boat using the sine or cosine function?
4. What would be the vertical distance to the boat if the observer moves halfway down the cliff?
5. How would you find the boat’s distance if it moves closer to the cliff?

**Tip:** The angle of depression is always measured from the horizontal, not the vertical.

From the top of a vertical cliff 125 m high, the angle of depression of a boat is 18.833°. Find the distance of the boat from the cliff.

Learn how to calculate the horizontal distance of a boat from the base of a 125m high cliff, given an angle of depression of 18.833°. This problem uses key trigonometric concepts, including the tangent function. Perfect for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation