A tree is broken by the wind. The top struck the ground at an angle of `45^(@)` and at a distance of 30 m from the root.
(i) Find whole height of the tree.
(ii) which mathematical concept is used in this problem ?
(iii) Which value is being emphasised here ?

Answer

To solve the problem step by step, we will break it down into parts as follows:

### Step 1: Understand the Problem
We have a tree that has broken and the top of the tree has struck the ground at a distance of 30 meters from the root, forming an angle of 45 degrees with the ground.

### Step 2: Identify the Triangle
We can visualize the situation as a right triangle where:
- The height of the broken part of the tree (vertical part) is represented as BC.
- The distance from the base of the tree to the point where the top of the tree touches the ground (horizontal part) is represented as AB, which is given as 30 m.
- The total height of the tree (the original height before it broke) is represented as AC.

### Step 3: Apply Trigonometric Ratios
Since we have a right triangle and we know one angle (45 degrees), we can use the tangent function:
\[ \tan(45^\circ) = \frac{BC}{AB} \]
Given that \(AB = 30\) m and \(\tan(45^\circ) = 1\):
\[ 1 = \frac{BC}{30} \]
Cross-multiplying gives:
\[ BC = 30 \text{ m} \]

### Step 4: Find the Length of AC
Next, we can use the sine function to find the total height of the tree (AC):
\[ \sin(45^\circ) = \frac{BC}{AC} \]
Substituting the known values:
\[ \frac{1}{\sqrt{2}} = \frac{30}{AC} \]
Cross-multiplying gives:
\[ AC = 30 \sqrt{2} \text{ m} \]
Calculating \(30 \sqrt{2}\):
\[ \sqrt{2} \approx 1.414 \implies AC \approx 30 \times 1.414 \approx 42.42 \text{ m} \]

### Step 5: Calculate the Whole Height of the Tree
The total height of the tree is the sum of BC and AC:
\[ \text{Total Height} = AC + BC = 42.42 \text{ m} + 30 \text{ m} = 72.42 \text{ m} \]

### Summary of Answers
(i) The whole height of the tree is approximately **72.42 meters**.

(ii) The mathematical concept used in this problem is **Trigonometry**.

(iii) The value being emphasized here is the importance of **environmental awareness**, particularly regarding deforestation and its consequences.