A tower is 50 m high. When the sun's altitude is `45^(@)` then what will be the length of its shadow ?

Answer

To solve the problem of finding the length of the shadow of a tower that is 50 m high when the sun's altitude is 45°, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a tower that is 50 m tall, and we need to find the length of its shadow when the sun's altitude is 45°.

2. **Drawing a Diagram**:
Draw a right triangle where:
- The height of the tower (AB) is 50 m (perpendicular side).
- The length of the shadow (BC) is what we need to find (base).
- The angle of elevation from the tip of the shadow to the top of the tower (angle C) is 45°.

3. **Using Trigonometric Ratios**:
We can use the tangent function, which relates the angle of elevation to the opposite side (height of the tower) and the adjacent side (length of the shadow).
\[ \tan(C) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
Here, \( AB = 50 \) m and \( BC = x \) (the length of the shadow).

4. **Substituting Known Values**:
Since angle C is 45°, we have:
\[ \tan(45°) = \frac{50}{x} \]

5. **Using the Value of Tan(45°)**:
We know that:
\[ \tan(45°) = 1 \]
Therefore, we can write:
\[ 1 = \frac{50}{x} \]

6. **Solving for x**:
Rearranging the equation gives us:
\[ x = 50 \]
Thus, the length of the shadow is 50 m.

### Final Answer:
The length of the shadow of the tower is **50 meters**.

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