To find the height of the tower, let the height of the tower be $h$ and the base of the tower be at point $O$. The points of observation are at distances $a$ and $b$ from the base of the tower. Using trigonometric relations, we apply the tangent function for both angles of elevation.

Step 1: Tan relations for the two points

From the point at distance $a$ where the angle of elevation is $30^\circ$: $\tan 30^\circ = \frac{h}{a}$ $\frac{1}{\sqrt{3}} = \frac{h}{a}$ $h = \frac{a}{\sqrt{3}}$

From the point at distance $b$ where the angle of elevation is $60^\circ$: $\tan 60^\circ = \frac{h}{b}$ $\sqrt{3} = \frac{h}{b}$ $h = b\sqrt{3}$

Step 2: Equating the height expressions

From the two expressions for $h$, we have: $\frac{a}{\sqrt{3}} = b\sqrt{3}$

Step 3: Solve for $b$

Multiply through by $\sqrt{3}$: $a = 3b$

Step 4: Substitute back to find $h$

From $h = b\sqrt{3}$ and $a = 3b$, substitute $b = \frac{a}{3}$: $h = \sqrt{3} \cdot \frac{a}{3}$ $h = \frac{a\sqrt{3}}{3}$

Final Answer:

The height of the tower is: $\boxed{\frac{a\sqrt{3}}{3}}$

Would you like more details or have any questions? Here are 5 related questions to consider:

1. How does the tangent function relate to the angle of elevation in such problems?
2. What would the solution look like if the angles were 45° and 30° instead?
3. Can we derive a general formula for such problems with arbitrary angles of elevation?
4. What happens if the points are not on the same straight line but at different distances?
5. How does this calculation change if the distances $a$ and $b$ are in different units?

Tip: Always double-check whether the angles are complementary (e.g., 30° and 60°) to simplify trigonometric calculations.