The shadow of a vertical tower on level ground increases by 10 m when the altitude of the sun changes from `45^(@)` to `30^(@)`. Find the height of the tower.

Answer

To find the height of the tower, we can use the information given about the angles of elevation and the change in the length of the shadow. Let's break down the solution step by step.

### Step 1: Define the Variables
Let:
- \( y \) = height of the tower
- \( x \) = length of the shadow when the angle of elevation is \( 45^\circ \)

### Step 2: Use the Tangent Function for the First Angle
For the angle of elevation of \( 45^\circ \):
\[ \tan(45^\circ) = \frac{\text{height of tower}}{\text{length of shadow}} = \frac{y}{x} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{y}{x} \implies y = x \quad \text{(Equation 1)} \]

### Step 3: Use the Tangent Function for the Second Angle
When the angle of elevation changes to \( 30^\circ \), the length of the shadow increases by 10 m, so the new length of the shadow is \( x + 10 \):
\[ \tan(30^\circ) = \frac{y}{x + 10} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{y}{x + 10} \]
Cross-multiplying gives:
\[ y = \frac{1}{\sqrt{3}}(x + 10) \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we have \( y = x \). Substitute \( y \) in Equation 2:
\[ x = \frac{1}{\sqrt{3}}(x + 10) \]

### Step 5: Solve for \( x \)
Multiply both sides by \( \sqrt{3} \) to eliminate the fraction:
\[ \sqrt{3}x = x + 10 \]
Rearranging gives:
\[ \sqrt{3}x - x = 10 \]
\[ (\sqrt{3} - 1)x = 10 \]
\[ x = \frac{10}{\sqrt{3} - 1} \]

### Step 6: Find \( y \)
Using Equation 1, \( y = x \):
\[ y = \frac{10}{\sqrt{3} - 1} \]

### Step 7: Rationalize the Denominator
To simplify \( y \):
\[ y = \frac{10(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{10(\sqrt{3} + 1)}{3 - 1} = \frac{10(\sqrt{3} + 1)}{2} = 5(\sqrt{3} + 1) \]

### Final Answer
Thus, the height of the tower is:
\[ y = 5(\sqrt{3} + 1) \text{ meters} \]