https://file.aiquickdraw.com/mathbot/file/1727382491080w1k6nujf.jpg

This problem involves calculating the horizontal distance between a boat and the base of the cliffs using trigonometry. Here's a step-by-step breakdown:

### Problem Recap:
- The cliffs are **90 m** high.
- Natalie, standing on top of the cliff, is **1.4 m** tall.
- The angle of depression to the boat from her line of sight is **39°**.

### Key Information:
- The total height from Natalie’s eye level to the water is:  
  \[
  \text{Total height} = 90 \, \text{m} + 1.4 \, \text{m} = 91.4 \, \text{m}.
  \]
  
- The **angle of depression** (39°) is equivalent to the angle between the boat, her line of sight, and the horizontal line (i.e., the angle of elevation at the boat).

### Step 1: Set up the trigonometric relation
- We want to find the horizontal distance (\(d\)) between the boat and the base of the cliff.
- Use the **tangent** function, which relates the opposite side (height) to the adjacent side (horizontal distance):
  \[
  \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}.
  \]
  
  Here, \(\theta = 39^\circ\) and the opposite side is the total height (91.4 m). Thus, the equation becomes:
  \[
  \tan(39^\circ) = \frac{91.4}{d}.
  \]

### Step 2: Solve for \(d\)
Rearranging the equation to find \(d\):
\[
d = \frac{91.4}{\tan(39^\circ)}.
\]

Now, I'll calculate the value.The horizontal distance between the boat and the base of the cliffs is approximately **112.9 meters**.

Would you like further details or clarifications? Here are five follow-up questions to deepen understanding:

1. How would the calculation change if Natalie were standing at a different height?
2. Can we use a different trigonometric function for this problem? Why or why not?
3. What if the angle of depression increased to 45°? What would the new distance be?
4. How does the angle of depression affect the distance calculation in general?
5. What is the significance of the angle of depression in real-world scenarios?

**Tip**: Always ensure to identify the right trigonometric function based on the given angle and sides of the triangle in height-distance problems.

The Cathedral Bluffs in Toronto, Ontario, are eroded sandstone cliffs that rise 90 m above Lake Ontario. Natalie is 1.4 m tall. From her position at the top of the cliffs, the angle between the surface of the lake and her line of sight to a boat is 39°. Find the distance between the boat and the base of the cliffs to the nearest tenth of a metre.

Calculate the horizontal distance from a boat to the base of the Cathedral Bluffs using trigonometry. Learn how to use the tangent function and the angle of depression to solve this real-world problem involving a 39° angle and a height of 91.4 meters. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation