A vertical tower standing on a leveled field is mounted with a vertical flag staff of length 3 m. From a point on the field, the angles of elevation of the bottom and tip of the flag staff are `30^(@) and 45^(@)` respectively, which one of the following gives the best approximation to the height of the tower?

Options

A. A) `3. 90 m `
B. B) `4.00 m`
C. C) `4. 10 m`
D. D) `4. 25 m`

Answer

To find the height of the tower, we can follow these steps:

### Step 1: Understand the Problem
We have a vertical tower with a height \( H \) meters, and a flagstaff of height 3 meters on top of the tower. The angles of elevation from a point on the ground to the bottom of the tower and the top of the flagstaff are given as \( 30^\circ \) and \( 45^\circ \) respectively.

### Step 2: Set Up the Diagram
Let:
- Point A be the point on the ground from where the angles are measured.
- Point B be the bottom of the tower.
- Point C be the top of the tower.
- Point D be the top of the flagstaff.

The height of the tower is \( H \) meters, and the height of the flagstaff is 3 meters. Thus, the total height from the ground to the top of the flagstaff (point D) is \( H + 3 \) meters.

### Step 3: Use Trigonometry for the Bottom of the Tower
From point A to point B (the bottom of the tower), the angle of elevation is \( 30^\circ \).
Using the tangent function:
\[ \tan(30^\circ) = \frac{H}{x} \]
Where \( x \) is the horizontal distance from point A to the base of the tower. We know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
So, we can write:
\[ \frac{1}{\sqrt{3}} = \frac{H}{x} \]
This implies:
\[ x = H \sqrt{3} \]

### Step 4: Use Trigonometry for the Top of the Flagstaff
From point A to point D (the top of the flagstaff), the angle of elevation is \( 45^\circ \).
Using the tangent function again:
\[ \tan(45^\circ) = \frac{H + 3}{x} \]
We know that:
\[ \tan(45^\circ) = 1 \]
So, we can write:
\[ 1 = \frac{H + 3}{x} \]
This implies:
\[ x = H + 3 \]

### Step 5: Set the Two Expressions for \( x \) Equal
Now we have two expressions for \( x \):
1. \( x = H \sqrt{3} \)
2. \( x = H + 3 \)

Setting them equal:
\[ H \sqrt{3} = H + 3 \]

### Step 6: Solve for \( H \)
Rearranging the equation:
\[ H \sqrt{3} - H = 3 \]
Factoring out \( H \):
\[ H(\sqrt{3} - 1) = 3 \]
So we can solve for \( H \):
\[ H = \frac{3}{\sqrt{3} - 1} \]

### Step 7: Rationalize the Denominator
To simplify \( H \), we can multiply the numerator and denominator by \( \sqrt{3} + 1 \):
\[ H = \frac{3(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{3(\sqrt{3} + 1)}{3 - 1} = \frac{3(\sqrt{3} + 1)}{2} \]

### Step 8: Approximate the Value of \( H \)
Calculating the approximate value:
\[ H \approx \frac{3(1.732 + 1)}{2} \approx \frac{3(2.732)}{2} \approx \frac{8.196}{2} \approx 4.098 \]
Thus, the best approximation for the height of the tower is approximately \( 4.1 \) meters.

### Final Answer
The best approximation to the height of the tower is \( 4.1 \) meters.

---