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 step by step, we will use trigonometric principles involving angles of depression and right triangles.

### Step 1: Understand the Problem
We have a tower of height 80 meters. From the top of the tower, the angles of depression to the top and bottom of a house are 30° and 45°, respectively. We need to find the height of the house.

### Step 2: Set Up the Diagram
1. Let \( A \) be the top of the tower, \( B \) be the bottom of the tower, \( C \) be the top of the house, and \( D \) be the bottom of the house.
2. The height of the tower \( AB = 80 \) meters.
3. Let the height of the house \( CD = h \) meters.
4. The distance from the tower to the house (horizontal distance) is \( D \).

### Step 3: Use the Angle of Depression to Find Distances
1. From point \( A \) (top of the tower), the angle of depression to point \( C \) (top of the house) is 30°.
- In triangle \( ABC \):
\[ \tan(30°) = \frac{BC}{AB} = \frac{D}{80 - h} \]
- Since \( \tan(30°) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{D}{80 - h} \implies D = \frac{(80 - h)}{\sqrt{3}} \quad \text{(Equation 1)} \]

2. From point \( A \), the angle of depression to point \( D \) (bottom of the house) is 45°.
- In triangle \( ABD \):
\[ \tan(45°) = \frac{BD}{AB} = \frac{D}{80} \]
- Since \( \tan(45°) = 1 \):
\[ 1 = \frac{D}{80} \implies D = 80 \quad \text{(Equation 2)} \]

### Step 4: Set Equations Equal
Now we have two expressions for \( D \):
1. From Equation 1: \( D = \frac{(80 - h)}{\sqrt{3}} \)
2. From Equation 2: \( D = 80 \)

Setting them equal gives:
\[ \frac{(80 - h)}{\sqrt{3}} = 80 \]

### Step 5: Solve for \( h \)
1. Cross-multiply:
\[ 80 - h = 80\sqrt{3} \]
2. Rearranging gives:
\[ h = 80 - 80\sqrt{3} \]

### Step 6: Simplify
1. Factor out \( 80 \):
\[ h = 80(1 - \sqrt{3}) \]

### Final Step: Calculate Height
The height of the house is:
\[ h = 80(1 - \sqrt{3}) \text{ meters} \]