A vertical flagstaff stands on a horizontal plane. From a point 100 m from its foot, the angle of elevation of its top is found to be `45^(@)`. Find the height of the flagstaff.

Answer

To find the height of the flagstaff, we can use the concept of trigonometry, specifically the tangent function. Here is the step-by-step solution:

### Step 1: Understand the problem
We have a vertical flagstaff and a point on the ground from where the angle of elevation to the top of the flagstaff is given. The distance from this point to the base of the flagstaff is also provided.

### Step 2: Draw a diagram
Draw a right triangle where:
- The vertical side represents the height of the flagstaff (let's denote it as \( h \)).
- The horizontal side represents the distance from the point to the base of the flagstaff, which is 100 m.
- The angle of elevation from the point to the top of the flagstaff is \( 45^\circ \).

### Step 3: Set up the tangent function
Using the definition of the tangent function in a right triangle:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
In our case, the angle \( \theta = 45^\circ \), the opposite side is the height of the flagstaff \( h \), and the adjacent side is the distance from the point to the base of the flagstaff, which is 100 m.

Thus, we can write:
\[ \tan(45^\circ) = \frac{h}{100} \]

### Step 4: Substitute the value of \( \tan(45^\circ) \)
We know that:
\[ \tan(45^\circ) = 1 \]
So we can substitute this into our equation:
\[ 1 = \frac{h}{100} \]

### Step 5: Solve for \( h \)
To find the height \( h \), we multiply both sides of the equation by 100:
\[ h = 1 \times 100 \]
\[ h = 100 \text{ m} \]

### Conclusion
The height of the flagstaff is \( 100 \) meters.

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