https://file.aiquickdraw.com/mathbot/file/1729231236904k67k79e3.jpg

To solve the questions presented in the image, we can apply trigonometry principles, specifically those involving angles of elevation.

### Given Information:
- Akshat observes a tree on the opposite bank of a river.
- Initial angle of elevation: 60°.
- After moving 30 meters away from the riverbank, the angle of elevation reduces to 30°.

### Let’s Define the Variables:
- Let **\( h \)** be the height of the tree.
- Let **\( x \)** be the width of the river.

### Step-by-step Solution:

1. **For the initial observation (60° angle of elevation):**

Using the tangent of the angle of elevation:

\[
\tan(60^\circ) = \frac{h}{x}
\]

We know that:

\[
\tan(60^\circ) = \sqrt{3}
\]

Thus, the equation becomes:

\[
\sqrt{3} = \frac{h}{x} \quad \text{(Equation 1)}
\]

From this, we can express **\( h \)** in terms of **\( x \)**:

\[
h = x\sqrt{3}
\]

2. **For the second observation (30° angle of elevation after moving 30 meters back):**

The distance from the observer to the tree is now **\( x + 30 \)**. Again using the tangent of the angle of elevation:

\[
\tan(30^\circ) = \frac{h}{x + 30}
\]

We know that:

\[
\tan(30^\circ) = \frac{1}{\sqrt{3}}
\]

Thus, the equation becomes:

\[
\frac{1}{\sqrt{3}} = \frac{h}{x + 30} \quad \text{(Equation 2)}
\]

3. **Solving the System of Equations:**

From Equation 1, we know that **\( h = x\sqrt{3} \)**. Substituting this into Equation 2:

\[
\frac{1}{\sqrt{3}} = \frac{x\sqrt{3}}{x + 30}
\]

Multiplying both sides by **\( \sqrt{3} \)**:

\[
1 = \frac{3x}{x + 30}
\]

Cross-multiply to solve for **\( x \)**:

\[
x + 30 = 3x
\]

\[
2x = 30
\]

\[
x = 15 \, \text{meters}
\]

So, the width of the river is **15 meters**.

4. **Height of the Tree:**

Using **\( x = 15 \)** in Equation 1:

\[
h = 15\sqrt{3} \approx 25.98 \, \text{meters}
\]

### Final Answers:
1. **Height of the tree**: ~25.98 meters.
2. **Width of the river**: 15 meters.

Would you like details on the remaining questions, or should I proceed further with the alternate scenario (Question 4)? Feel free to ask!

---

Here are 5 related questions to further your understanding:

1. How does trigonometry help in measuring heights and distances?
2. What are the key trigonometric ratios and their real-world applications?
3. How can you use similar triangles to estimate distances in this scenario?
4. Why does the angle of elevation change as the observer moves farther?
5. How do we derive values like \( \tan(60^\circ) \) and \( \tan(30^\circ) \)?

**Tip:** Always double-check the units (meters, degrees) and conversions when solving word problems to avoid mistakes!

Akshat observes a tree on the opposite bank of a river. The angle of elevation from the bank to the top of the tree is 60°. After moving 30 meters away from the river, the angle of elevation reduces to 30°. Find the height of the tree and the width of the river.

Learn how to calculate the height of a tree and the width of a river using angles of elevation and trigonometry. This problem involves a right triangle, tangent functions, and real-world applications of trigonometric ratios.

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