The shadow of a vertical tower becomes 30 metres longer when the altitude of the sun changes from `60^(@)"to"45^(@)` .find the height of the tower .

Options

A. `15sqrt(3)` metres
B. `15(3+sqrt(3))` metres
C. `15(3-sqrt(3))` metres
D. `12(3+sqrt(3))` metres

Answer

To solve the problem, we need to find the height of the tower based on the information given about the shadow lengths at different angles of elevation of the sun.

### Step 1: Define Variables
Let \( h \) be the height of the tower in meters.

### Step 2: Use Trigonometry for the First Angle
When the angle of elevation of the sun is \( 60^\circ \), we can use the tangent function:
\[ \tan(60^\circ) = \frac{h}{s_1} \]
where \( s_1 \) is the length of the shadow when the angle is \( 60^\circ \). We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, we have:
\[ \sqrt{3} = \frac{h}{s_1} \implies s_1 = \frac{h}{\sqrt{3}} \]

### Step 3: Use Trigonometry for the Second Angle
When the angle of elevation of the sun changes to \( 45^\circ \), we can again use the tangent function:
\[ \tan(45^\circ) = \frac{h}{s_2} \]
where \( s_2 \) is the length of the shadow when the angle is \( 45^\circ \). We know that:
\[ \tan(45^\circ) = 1 \]
Thus, we have:
\[ 1 = \frac{h}{s_2} \implies s_2 = h \]

### Step 4: Set Up the Equation
According to the problem, the shadow becomes 30 meters longer when the angle changes from \( 60^\circ \) to \( 45^\circ \). Therefore, we can write:
\[ s_2 = s_1 + 30 \]
Substituting the expressions for \( s_1 \) and \( s_2 \):
\[ h = \frac{h}{\sqrt{3}} + 30 \]

### Step 5: Solve for \( h \)
To eliminate the fraction, we can multiply the entire equation by \( \sqrt{3} \):
\[ \sqrt{3}h = h + 30\sqrt{3} \]
Now, rearranging the equation gives:
\[ \sqrt{3}h - h = 30\sqrt{3} \]
Factoring out \( h \):
\[ h(\sqrt{3} - 1) = 30\sqrt{3} \]
Now, we can solve for \( h \):
\[ h = \frac{30\sqrt{3}}{\sqrt{3} - 1} \]

### Step 6: Rationalize the Denominator
To simplify further, we can multiply the numerator and denominator by \( \sqrt{3} + 1 \):
\[ h = \frac{30\sqrt{3}(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{30\sqrt{3}(\sqrt{3} + 1)}{3 - 1} = \frac{30\sqrt{3}(\sqrt{3} + 1)}{2} \]
This simplifies to:
\[ h = 15\sqrt{3}(\sqrt{3} + 1) = 15(3 + \sqrt{3}) = 45 + 15\sqrt{3} \]

### Final Answer
Thus, the height of the tower is:
\[ h = 45 + 15\sqrt{3} \text{ meters} \]