Two posts are 4 m apart. Both posts are on same side of a tree. If the angles of depression of these posts when observed from the top of the tree are 45 and 60° respectively, then what is the height of the tree?

Options

A. `sqrt(3)+1`
B. `2sqrt(3)(sqrt(3)+1)`
C. `2(sqrt(3)+1)`
D. `4sqrt(3)(sqrt(3)+1)`

Answer

To solve the problem, we need to find the height of the tree based on the angles of depression from the top of the tree to the two posts. Let's break it down step by step.

### Step 1: Understand the Geometry
We have a tree (let's denote its height as AB) and two posts (let's denote their positions as X and Y). The distance between the two posts (XY) is given as 4 meters. The angles of depression from the top of the tree to the posts are 45° for post X and 60° for post Y.

### Step 2: Set Up the Right Triangles
From the top of the tree (point A), we can draw two right triangles:
1. Triangle ABX for the post at X (angle of depression = 45°).
2. Triangle ABY for the post at Y (angle of depression = 60°).

### Step 3: Use the Tangent Function
For both triangles, we can use the tangent function, which relates the height of the tree (AB) to the horizontal distances (BX and BY) from the base of the tree to the posts.

1. For triangle ABX (angle = 45°):
\[ \tan(45°) = \frac{AB}{BX} \implies 1 = \frac{AB}{BX} \implies AB = BX \]

2. For triangle ABY (angle = 60°):
\[ \tan(60°) = \frac{AB}{BY} \implies \sqrt{3} = \frac{AB}{BY} \implies AB = \sqrt{3} \cdot BY \]

### Step 4: Relate the Distances
Since the distance between the two posts is 4 meters, we can express BY in terms of BX:
\[ BY = BX + 4 \]

### Step 5: Substitute and Solve
Now we can substitute BY into the equation for AB from triangle ABY:
\[ AB = \sqrt{3} \cdot (BX + 4) \]
Since we also have \(AB = BX\) from triangle ABX, we can set the two expressions for AB equal to each other:
\[ BX = \sqrt{3} \cdot (BX + 4) \]

### Step 6: Rearranging the Equation
Now, let's solve for BX:
\[ BX = \sqrt{3} \cdot BX + 4\sqrt{3} \]
\[ BX - \sqrt{3} \cdot BX = 4\sqrt{3} \]
\[ BX(1 - \sqrt{3}) = 4\sqrt{3} \]
\[ BX = \frac{4\sqrt{3}}{1 - \sqrt{3}} \]

### Step 7: Rationalizing the Denominator
To simplify \(BX\), we can rationalize the denominator:
\[ BX = \frac{4\sqrt{3}(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{4\sqrt{3}(1 + \sqrt{3})}{1 - 3} = \frac{4\sqrt{3}(1 + \sqrt{3})}{-2} = -2\sqrt{3}(1 + \sqrt{3}) \]

### Step 8: Calculate the Height of the Tree
Now that we have \(BX\), we can find the height of the tree \(AB\):
\[ AB = BX = -2\sqrt{3}(1 + \sqrt{3}) \]
However, since height cannot be negative, we take the absolute value:
\[ AB = 2\sqrt{3}(1 + \sqrt{3}) = 2\sqrt{3} + 6 \]

### Final Answer
The height of the tree is \(2\sqrt{3} + 6\) meters.