The angle of elevation of the top of a flag post from a point 5 m away from its base is `75^(@).` What is the approximate height of the flag post?

Options

A. 15 m
B. 17 m
C. 19 m
D. 21 m

Answer

To solve the problem of finding the height of the flag post given the angle of elevation and the distance from the base, we can follow these steps:

### Step 1: Understand the Problem
We have a flag post (AB) and a point (C) that is 5 meters away from the base (A) of the flag post. The angle of elevation from point C to the top of the flag post (B) is given as 75 degrees.

### Step 2: Set Up the Triangle
In triangle ABC:
- AB is the height of the flag post (h).
- AC is the distance from the point to the base of the flag post, which is 5 m.
- Angle CAB (the angle of elevation) is 75 degrees.

### Step 3: Use the Tangent Function
We can use the tangent function to relate the height of the flag post to the distance from the base:
\[ \tan(75^\circ) = \frac{AB}{AC} = \frac{h}{5} \]

### Step 4: Solve for h
Rearranging the equation gives us:
\[ h = 5 \cdot \tan(75^\circ) \]

### Step 5: Calculate \(\tan(75^\circ)\)
Using a calculator or trigonometric tables, we find:
\[ \tan(75^\circ) \approx 3.732 \]

### Step 6: Substitute and Calculate h
Now substituting the value of \(\tan(75^\circ)\) into the equation:
\[ h = 5 \cdot 3.732 \approx 18.66 \text{ meters} \]

### Step 7: Round the Result
Rounding 18.66 to the nearest whole number gives us:
\[ h \approx 19 \text{ meters} \]

### Final Answer
The approximate height of the flag post is **19 meters**.

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