The angle of elevation of the top of a tower from the top and bottom of a building of height 'a' are `30^@` and `45^@` respectively. If the tower and the building stand at the same level , the height of the tower is

Options

A. `(a(3+sqrt3))/2`
B. `a(sqrt3+1)`
C. `sqrt3a`
D. `a(sqrt3-1)`

Explanation

To solve the problem step by step, we will denote the height of the tower as \( h \) and the height of the building as \( a \). We will also denote the distance from the base of the building to the base of the tower as \( d \).

### Step 1: Set up the problem
From the problem, we know:
- The angle of elevation from the top of the building to the top of the tower is \( 30^\circ \).
- The angle of elevation from the bottom of the building to the top of the tower is \( 45^\circ \).

### Step 2: Use the tangent function for the angles
Using the tangent function, we can set up the following equations based on the angles of elevation:

1. From the top of the building (height \( a \)):
\[ \tan(30^\circ) = \frac{h - a}{d} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we can rewrite this as:
\[ \frac{1}{\sqrt{3}} = \frac{h - a}{d} \implies d = \sqrt{3}(h - a) \quad \text{(Equation 1)} \]

2. From the bottom of the building (height \( 0 \)):
\[ \tan(45^\circ) = \frac{h}{d} \]
Since \( \tan(45^\circ) = 1 \), we can rewrite this as:
\[ 1 = \frac{h}{d} \implies d = h \quad \text{(Equation 2)} \]

### Step 3: Equate the two expressions for \( d \)
From Equation 1 and Equation 2, we have:
\[ \sqrt{3}(h - a) = h \]

### Step 4: Solve for \( h \)
Now, we can solve for \( h \):
\[ \sqrt{3}h - \sqrt{3}a = h \]
Rearranging gives:
\[ \sqrt{3}h - h = \sqrt{3}a \]
Factoring out \( h \):
\[ h(\sqrt{3} - 1) = \sqrt{3}a \]
Thus, we can solve for \( h \):
\[ h = \frac{\sqrt{3}a}{\sqrt{3} - 1} \]

### Step 5: Rationalize the denominator
To simplify further, we can rationalize the denominator:
\[ h = \frac{\sqrt{3}a(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{\sqrt{3}a(\sqrt{3} + 1)}{3 - 1} = \frac{\sqrt{3}a(\sqrt{3} + 1)}{2} \]

### Final Answer
Thus, the height of the tower is:
\[ h = \frac{\sqrt{3}a(\sqrt{3} + 1)}{2} \]