The angle of elevation of the top of am incomplete temple, at a point 30 m away from its foot, is `30^(@)`. How much more high the temple must ne constructed so that the angle of elevation of its top at the same point be `45^(@)`.

Answer

To solve the problem step by step, we will use trigonometric principles, specifically the tangent function, which relates the angles of elevation to the heights and distances involved.

### Step 1: Understand the Problem
We have a temple with an incomplete height. We need to find out how much more height is required so that the angle of elevation from a point 30 meters away from the foot of the temple becomes 45 degrees.

### Step 2: Set Up the Diagram
Let:
- Point A be the foot of the temple.
- Point B be the top of the incomplete temple.
- Point C be the point from which the angle of elevation is measured, which is 30 meters away from A.

### Step 3: Use the First Angle of Elevation (30 degrees)
From point C, the angle of elevation to point B is 30 degrees. We can use the tangent function:
\[ \tan(30^\circ) = \frac{AB}{BC} \]
Where:
- \( AB \) is the height of the incomplete temple.
- \( BC = 30 \) m (the distance from the point to the foot of the temple).

Using the value of \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{AB}{30} \]
From this, we can find \( AB \):
\[ AB = 30 \cdot \frac{1}{\sqrt{3}} = \frac{30}{\sqrt{3}} \approx 10\sqrt{3} \text{ m} \]

### Step 4: Set Up the Second Angle of Elevation (45 degrees)
Now, we want the angle of elevation to be 45 degrees when the temple is constructed to a new height \( PB \). Let \( x \) be the additional height needed. Thus, the new height \( PB \) will be:
\[ PB = AB + x \]
Using the tangent function again:
\[ \tan(45^\circ) = \frac{PB}{BC} \]
Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{AB + x}{30} \]
This simplifies to:
\[ AB + x = 30 \]

### Step 5: Substitute the Value of AB
Now, substitute \( AB \) into the equation:
\[ \frac{30}{\sqrt{3}} + x = 30 \]
To isolate \( x \):
\[ x = 30 - \frac{30}{\sqrt{3}} \]

### Step 6: Simplify the Expression for x
To simplify \( x \):
\[ x = 30 \left(1 - \frac{1}{\sqrt{3}}\right) \]
Calculating \( \frac{1}{\sqrt{3}} \approx 0.577 \):
\[ x \approx 30 \cdot (1 - 0.577) \approx 30 \cdot 0.423 \approx 12.68 \text{ m} \]

### Final Answer
The additional height required to construct the temple is approximately **12.68 meters**.

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