A vertical pole and a vertical tower are on the same level ground in such a way that, from the top of the pole, the angle of elevation of the top of the tower is 60° and the angle of depression of the bottom of the tower is 30°. If the height of the pole is 24 m, then find the height of the tower (in m).

The correct answer is 96.

To find the height of the tower, we can represent the setup using right-angled triangles based on the given angles of elevation and depression.

Step 1: Visualize the problem and label the components
Let $AB$ represent the vertical pole of height 24 m, where $A$ is the top and $B$ is the bottom of the pole.
Let $CD$ represent the vertical tower on the same level ground, where $C$ is the top and $D$ is the bottom of the tower.
Let $x$ be the horizontal distance between the pole and the tower, so $BD=x$.
Draw a horizontal line from the top of the pole $A$ to meet the tower $CD$ perpendicularly at point $E$. Therefore, $AE=BD=x$ and $ED=AB=24\text{m}$.

The height of the tower is $CD=CE+ED=CE+24$.

Step 2: Find the horizontal distance using the angle of depression
From the top of the pole $A$, the angle of depression of the bottom of the tower $D$ is 30°. This corresponds to the angle in the right-angled triangle $AED$ where $\angle EAD=30°$.
Using the trigonometric ratio for tangent in triangle $AED$:

$\tan (30°)=\frac{ED}{AE}$

We know that $ED=24\text{m}$, $AE=x$, and $\tan (30°)=\frac{1}{\sqrt{3}}$. Substituting these values:

$\frac{1}{\sqrt{3}}=\frac{24}{x}$

Solving for $x$:

$x=24\sqrt{3}\text{m}$

Step 3: Find the upper segment of the tower using the angle of elevation
From the top of the pole $A$, the angle of elevation of the top of the tower $C$ is 60°. This corresponds to the angle in the right-angled triangle $AEC$ where $\angle EAC=60°$.
Using the trigonometric ratio for tangent in triangle $AEC$:

$\tan (60°)=\frac{CE}{AE}$

We know that $AE=x=24\sqrt{3}\text{m}$ and $\tan (60°)=\sqrt{3}$. Substituting these values:

$\sqrt{3}=\frac{CE}{24\sqrt{3}}$

Solving for $CE$:

$CE=24\sqrt{3}\times \sqrt{3}$

$CE=24\times 3=72\text{m}$

Step 4: Calculate the total height of the tower
Now, we sum the two vertical parts of the tower to find its total height:

$\text{Height of the tower}\left(CD\right)=CE+ED$

$CD=72+24=96\text{m}$

Thus, the total height of the tower is 96 meters.