A tree Ac is broken over by wind from B . D is the point where the top of the broken tree touches the ground and BD makes an angle of `45^(@)` with the ground . If the distance between the base of the tree and the point D = 10 m What is the height of the tree ?

Options

A. `20 sqrt(3) m `
B. 60 m
C. `20 ( sqrt(3) - 1) m `
D. `40 sqrt(3) m `

Answer

To solve the problem, we need to find the height of the tree given the conditions described. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
- We have a tree that is broken at point B, and the top of the tree touches the ground at point D.
- The distance from the base of the tree (point A) to point D is given as 10 m.
- The segment BD makes an angle of 45 degrees with the ground.

### Step 2: Set Up the Right Triangle
- In triangle ABD, where:
- AB is the height of the tree (which we need to find),
- BD is the length of the broken part of the tree,
- AD is the horizontal distance from the base of the tree to point D (which is 10 m).

### Step 3: Use Trigonometric Ratios
- Since angle BDA is 45 degrees, we can use the tangent function:
\[ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{AD} \]
- We know that \(\tan(45^\circ) = 1\), so:
\[ 1 = \frac{AB}{10} \]
- This implies that:
\[ AB = 10 \text{ m} \]

### Step 4: Find the Length of BD
- Since BD makes a 45-degree angle with the ground, we can also use the sine function:
\[ \sin(45^\circ) = \frac{AB}{BD} \]
- We know that \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\), so:
\[ \frac{1}{\sqrt{2}} = \frac{10}{BD} \]
- Rearranging gives us:
\[ BD = 10\sqrt{2} \text{ m} \]

### Step 5: Calculate the Total Height of the Tree
- The total height of the tree (AC) is the sum of AB and BD:
\[ AC = AB + BD = 10 + 10\sqrt{2} \]
- Simplifying gives:
\[ AC = 10(1 + \sqrt{2}) \text{ m} \]

### Final Answer
The height of the tree is:
\[ \text{Height of the tree} = 10(1 + \sqrt{2}) \text{ m} \]