A ladder `6 m` long reaches a point `6 m` below the top of a vertical flagstaff. From the foot of the ladder, the elevation of the top of the flagstaff is `75^(@)`. What is the height of the flagstaff?

Options

A. `12 m `
B. `9 m`
C. `(6+sqrt(3)) m `
D. `(6+3sqrt(3)) m`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the problem
We have a ladder of length 6 m that reaches a point 6 m below the top of a vertical flagstaff. The angle of elevation from the foot of the ladder to the top of the flagstaff is 75 degrees. We need to find the height of the flagstaff.

### Step 2: Define the variables
Let:
- \( AC \) = height of the flagstaff
- \( AB \) = height from the top of the flagstaff to the point where the ladder touches the flagstaff (which is 6 m below the top)
- \( BC \) = the height from the point where the ladder touches the flagstaff to the ground

From the problem, we know:
- \( AB = 6 \, m \)
- The length of the ladder \( AD = 6 \, m \)

### Step 3: Set up the triangle
From the foot of the ladder (point \( D \)), the angle of elevation to the top of the flagstaff (point \( A \)) is \( 75^\circ \). We can visualize the situation as a right triangle \( ABD \) where:
- \( AD \) is the hypotenuse (the ladder)
- \( AB \) is the vertical leg (height from the top of the flagstaff to the point where the ladder touches)
- \( BD \) is the horizontal leg (distance from the foot of the ladder to the base of the flagstaff)

### Step 4: Use trigonometric ratios
In triangle \( ABD \):
- We can use the sine function:
\[ \sin(75^\circ) = \frac{AB}{AD} \]
Substituting the known values:
\[ \sin(75^\circ) = \frac{6}{6} \]
This simplifies to:
\[ \sin(75^\circ) = 1 \]
This indicates that the height \( AB \) is indeed equal to the length of the ladder.

### Step 5: Find \( BC \)
Now, we need to find \( BC \) using the cosine function:
\[ \cos(75^\circ) = \frac{BD}{AD} \]
Rearranging gives us:
\[ BD = AD \cdot \cos(75^\circ) = 6 \cdot \cos(75^\circ) \]

### Step 6: Calculate \( BC \)
Using the approximate value of \( \cos(75^\circ) \approx 0.2588 \):
\[ BD = 6 \cdot 0.2588 \approx 1.5528 \, m \]

### Step 7: Calculate the total height of the flagstaff \( AC \)
Now, we can find the total height of the flagstaff:
\[ AC = AB + BC = 6 + 1.5528 \approx 7.5528 \, m \]

### Final Answer
The height of the flagstaff is approximately \( 7.55 \, m \).