If from the top of a tower 80 meters high the angles of depression of the top and bottom of a house are `30^(@) and 45^(@)` respectively, then the height of the house is

Options

A. `40sqrt3` meters
B. `40((sqrt3-1)/(sqrt3))` meters
C. `80((sqrt3-1)/(sqrt3))` meters
D. `40((sqrt3+1)/(sqrt3))` meters

Answer

To solve the problem, we need to determine the height of the house based on the given angles of depression from the top of the tower. Let's break down the solution step by step.

### Step 1: Understand the Problem and Draw the Diagram
We have a tower AB that is 80 meters high. The angles of depression from the top of the tower (point A) to the top (point C) and bottom (point D) of the house are 30° and 45°, respectively.

### Step 2: Set Up the Diagram
- Let AB = 80 m (height of the tower).
- Let CD = h (height of the house).
- Let BD = x (distance from the base of the tower to the base of the house).
- The angle of depression to point C (top of the house) is 30°.
- The angle of depression to point D (bottom of the house) is 45°.

### Step 3: Use Trigonometric Ratios
1. For triangle ACD (using angle of depression to the top of the house):
- The angle of elevation from point C to point A is also 30°.
- Using the tangent function:
\[ \tan(30°) = \frac{AB - CD}{BD} = \frac{80 - h}{x} \]
- We know that \(\tan(30°) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{80 - h}{x} = \frac{1}{\sqrt{3}} \quad \text{(1)} \]

2. For triangle ABD (using angle of depression to the bottom of the house):
- The angle of elevation from point D to point A is also 45°.
- Using the tangent function:
\[ \tan(45°) = \frac{AB}{BD} = \frac{80}{x} \]
- We know that \(\tan(45°) = 1\), so:
\[ \frac{80}{x} = 1 \quad \Rightarrow \quad x = 80 \quad \text{(2)} \]

### Step 4: Substitute x into Equation (1)
Now we substitute the value of \(x\) from equation (2) into equation (1):
\[ \frac{80 - h}{80} = \frac{1}{\sqrt{3}} \]
Cross-multiplying gives:
\[ 80 - h = \frac{80}{\sqrt{3}} \]
Rearranging to solve for \(h\):
\[ h = 80 - \frac{80}{\sqrt{3}} \]

### Step 5: Simplify the Expression for h
To simplify:
\[ h = 80\left(1 - \frac{1}{\sqrt{3}}\right) \]
This can be rewritten as:
\[ h = 80\left(\frac{\sqrt{3} - 1}{\sqrt{3}}\right) \]

### Final Result
Thus, the height of the house \(CD\) is:
\[ h = \frac{80(\sqrt{3} - 1)}{\sqrt{3}} \text{ meters} \]