https://file.aiquickdraw.com/mathbot/file/1729231361548v3px6q8n.jpg

Let's analyze the problem step by step.

1. **Given**:
   - The observer is standing on one bank of a river, looking at a tree on the opposite bank.
   - Initially, the angle of elevation to the top of the tree is \(60^\circ\).
   - The observer moves 30 m away from the river, and the angle of elevation reduces to \(30^\circ\).

### To solve:
We can model this situation using trigonometry, particularly the concept of right triangles and the tangent function.

### **Question 1**: *What is the height of the tree?*
- Let \(h\) be the height of the tree, and let \(x\) be the width of the river (the distance from the observer to the base of the tree).
  
  Using the tangent function:
  \[
  \tan(60^\circ) = \frac{h}{x}
  \]
  \[
  h = x \cdot \tan(60^\circ) = x \cdot \sqrt{3} \quad \text{(since } \tan(60^\circ) = \sqrt{3}\text{)}
  \]

- After moving 30 m away from the river, the new distance from the observer to the tree is \(x + 30\). The new angle of elevation is \(30^\circ\), so:
  \[
  \tan(30^\circ) = \frac{h}{x + 30}
  \]
  \[
  \frac{1}{\sqrt{3}} = \frac{h}{x + 30}
  \]
  \[
  h = \frac{x + 30}{\sqrt{3}}
  \]

### **Now, equate the two expressions for \(h\):**
\[
x \sqrt{3} = \frac{x + 30}{\sqrt{3}}
\]
Multiply both sides by \(\sqrt{3}\):
\[
3x = x + 30
\]
\[
2x = 30
\]
\[
x = 15 \, \text{m}
\]
So, the width of the river is 15 m.

### Now, substitute \(x = 15\) into the first equation to find the height of the tree:
\[
h = 15 \cdot \sqrt{3} \approx 15 \cdot 1.732 = 25.98 \, \text{m}
\]
Thus, the height of the tree is approximately **25.98 m**.

---

### **Question 2**: *What is the width of the river?*
- From the previous calculations, the width of the river is **15 m**.

---

### **Question 3**: *After moving how much distance away from the river does the angle of elevation become 45°?*
- For the angle of elevation to be \(45^\circ\), the distance from the tree must satisfy:
  \[
  \tan(45^\circ) = \frac{h}{x + d}
  \]
  \[
  1 = \frac{h}{x + d} \quad \text{(since } \tan(45^\circ) = 1\text{)}
  \]
  \[
  h = x + d
  \]
  Substituting \(h = 25.98\) m and \(x = 15\) m:
  \[
  25.98 = 15 + d
  \]
  \[
  d = 25.98 - 15 = 10.98 \, \text{m}
  \]
  So, the observer must move **10.98 m** away for the angle to become \(45^\circ\).

---

### **Question 4**: *If the width of the river was 40 m, what would be the height of the tree?*
- If \(x = 40\) m, then using the first equation:
  \[
  h = 40 \cdot \tan(60^\circ) = 40 \cdot \sqrt{3} \approx 40 \cdot 1.732 = 69.28 \, \text{m}
  \]
  The height of the tree would be **69.28 m**.

---

Would you like more details on any of these solutions? Here are some related questions you might find helpful:

1. What if the observer had moved another 20 meters further from their final position?
2. How would the height of the tree change if the angles were \(45^\circ\) and \(15^\circ\)?
3. Can you solve this using the sine rule?
4. How can the observer use only the 30-meter movement and angles to estimate tree height directly?
5. What if the tree was on the same side of the river but at a different height?

**Tip:** In trigonometry, angles of \(30^\circ\), \(45^\circ\), and \(60^\circ\) often appear with simple trigonometric ratios, so it's helpful to memorize those!

Akshat studies in DAV Public school, Vasant Kunj, Delhi. During summer vacation he went to his native place in a village. His grandfather took him to the bank of a nearby river. Akshat was very happy to see the pollution-free environment near the river. He was standing on the bank of the river and thought to measure the width of the river. He found that the angle of elevation of the top of a tree standing on the opposite bank was 60°. When he moved 30 m away from the river, the angle of elevation reduced to 30°. What is the height of the tree? What is the width of the river? After moving how much distance away from the river does the angle of elevation become 45°, or if the width of the river were 40 m, what would be the height of the tree selected?

This problem involves calculating the height of a tree and the width of a river using angles of elevation and trigonometry. The observer moves 30 meters away from the river, and the angle changes from 60° to 30°. This exercise covers key trigonometric concepts suitable for 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