From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 25 m high building are $30^\circ$ and $60^\circ$ respectively. Find the height of the transmission tower. [Use $\sqrt{3} = 1\cdot732$]

Step 1: Define variables and set up the first triangle

Let $h$ be the height of the transmission tower and $x$ be the distance from the point on the ground to the base of the building. The building height is given as 25 m. We can use the tangent function, which relates the angle of elevation to the opposite side (height) and the adjacent side (distance from the building).

Step 2: Calculate distance to the building using the first angle

For the angle of elevation to the bottom of the tower (top of the building), which is $30^\circ$, the opposite side is the height of the building (25 m) and the adjacent side is $x$. We substitute the value of $\tan 30^\circ = \frac{1}{\sqrt{3}}$ to find $x$.

Step 3: Set up the second triangle for the top of the tower

Now, consider the angle of elevation to the top of the transmission tower, which is $60^\circ$. The total height from the ground to the top of the tower is the sum of the building's height and the tower's height, i.e., $25 + h$. The adjacent side remains $x$.

Step 4: Solve for the height of the tower

Substitute the value of $x = 25\sqrt{3}$ and $\tan 60^\circ = \sqrt{3}$ into the equation from the previous step. Solve the resulting equation for $h$.

Step 5: Final Answer

The height of the transmission tower is 50 meters. The value of $\sqrt{3}$ was provided but not needed for the exact answer, only if an approximate decimal answer was required.

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