Step 1: Define variables and set up right triangles

Let the distance from the point on the ground to the base of the pole be $x$. Let the height of the pole be $H = 18\text{ m}$ and the height of the flag be $h$.

$H = 18\text{ m}, \quad \theta_1 = 60°, \quad \theta_2 = 30°$

Step 2: Apply tangent for the bottom of the flag (pole height)

Using the angle of elevation to the bottom of the pole ($30°$): $\tan 30°= \frac{H}{x} = \frac{18}{x}$.

x = \frac{18}{\tan 30° = 18\sqrt{3}\text{ m}

Step 3: Apply tangent for the top of the flag

Using the angle of elevation to the top of the flag ($60°$): $\tan 60°= \frac{H + h}{x} = \frac{18 + h}{x}$.

$18 + h = x \tan 60°= (18\sqrt{3}) \times \sqrt{3} = 54\text{ m}$

Step 4: Calculate the height of the flag ($h$)

Subtract the pole height $H$ from the total height: $h = 54 - 18 = 36\text{ m}$.

$h = 36\text{ m}$