The angle of elevation of the top of a tower from the foot of a house, situated at a distance of 20 m from the tower is `60^(@)`. From the top of the top of the house the angle of elevation of the top of the tower os `45^(@)`. Find the height of house and tower.

Answer

To solve the problem, we need to find the height of the house and the height of the tower using the given angles of elevation and distances. Let's break it down step by step.

### Step 1: Understand the Problem
We have a tower and a house. The distance from the foot of the house to the tower is 20 meters. The angle of elevation from the foot of the house to the top of the tower is 60 degrees. From the top of the house, the angle of elevation to the top of the tower is 45 degrees. We need to find the heights of both the tower and the house.

### Step 2: Set Up the Diagram
- Let the height of the tower be \( T \).
- Let the height of the house be \( H \).
- The distance from the house to the tower is 20 meters.

### Step 3: Use Trigonometry to Find the Height of the Tower
From the foot of the house, we have:
- Angle of elevation to the top of the tower = 60 degrees
- Distance from the house to the tower = 20 meters

Using the tangent function:
\[ \tan(60^\circ) = \frac{T}{20} \]
We know that \( \tan(60^\circ) = \sqrt{3} \). Therefore:
\[ \sqrt{3} = \frac{T}{20} \]
Multiplying both sides by 20 gives:
\[ T = 20\sqrt{3} \]

Calculating \( T \):
\[ T = 20 \times 1.732 \approx 34.64 \text{ meters} \]

### Step 4: Use Trigonometry to Find the Height of the House
From the top of the house, we have:
- Angle of elevation to the top of the tower = 45 degrees
- The distance from the top of the house to the tower is still 20 meters.

Using the tangent function again:
\[ \tan(45^\circ) = \frac{T - H}{20} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{T - H}{20} \]
Multiplying both sides by 20 gives:
\[ T - H = 20 \]
Substituting \( T = 34.64 \):
\[ 34.64 - H = 20 \]
Solving for \( H \):
\[ H = 34.64 - 20 = 14.64 \text{ meters} \]

### Step 5: Final Heights
- Height of the tower, \( T \) = 34.64 meters
- Height of the house, \( H \) = 14.64 meters

### Summary
The height of the tower is approximately 34.64 meters, and the height of the house is approximately 14.64 meters.