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This problem involves two observations of the same tower from different points, where the angles of elevation are given. We can solve it using trigonometric principles.

Let the following be the setup:
- Let the height of the tower be \( h \).
- Let the horizontal distance from the first observation point to the base of the tower be \( d \).

From the information:
- The angle of elevation at the first point is \( 60^\circ \).
- The second observation point is 30 m vertically above the first point with an angle of elevation of \( 45^\circ \).

### Step 1: Apply Trigonometry to the First Point
From the first observation point, the angle of elevation is \( 60^\circ \), so using the tangent of the angle:
\[
\tan(60^\circ) = \frac{h}{d}
\]
Since \( \tan(60^\circ) = \sqrt{3} \), we get:
\[
\sqrt{3} = \frac{h}{d}
\]
Thus, 
\[
h = \sqrt{3} \cdot d \tag{1}
\]

### Step 2: Apply Trigonometry to the Second Point
The second point is 30 m vertically above the first, so the remaining height of the tower above the second point is \( h - 30 \). The angle of elevation at this point is \( 45^\circ \), and we know that \( \tan(45^\circ) = 1 \), giving:
\[
\tan(45^\circ) = \frac{h - 30}{d}
\]
Thus:
\[
1 = \frac{h - 30}{d} \quad \Rightarrow \quad d = h - 30 \tag{2}
\]

### Step 3: Solve the System of Equations
We now have two equations:
1. \( h = \sqrt{3} \cdot d \)
2. \( d = h - 30 \)

Substitute equation (1) into equation (2):
\[
d = \sqrt{3} \cdot d - 30
\]
This simplifies to:
\[
d(1 - \sqrt{3}) = -30
\]
Solving for \( d \):
\[
d = \frac{30}{\sqrt{3} - 1}
\]
We can rationalize the denominator by multiplying numerator and denominator by \( \sqrt{3} + 1 \):
\[
d = \frac{30 (\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{30 (\sqrt{3} + 1)}{2}
\]
Thus, the horizontal distance \( d \) is:
\[
d = 15 (\sqrt{3} + 1) \approx 41.96 \, \text{m}
\]

### Step 4: Find the Height of the Tower
Now substitute \( d = 15 (\sqrt{3} + 1) \) into equation (1) to find the height \( h \):
\[
h = \sqrt{3} \cdot 15 (\sqrt{3} + 1) = 15 (3 + \sqrt{3}) \approx 72.32 \, \text{m}
\]

### Final Answer:
- The height of the tower is approximately **72.32 m**.
- The horizontal distance from the first point is approximately **41.96 m**.

Do you have any questions or need further clarification?

Here are 5 related questions you might explore:
1. How would the solution change if the angle of elevation from the second point was different?
2. How can this be applied in real-life surveying of tall structures?
3. What happens if the two observation points are at different horizontal distances?
4. How would the solution be affected if the tower was not vertical?
5. Can you derive the general formula for any angles of elevation?

**Tip:** Always double-check your trigonometric values and units, especially when dealing with real-world measurements!

The angle of elevation of the top of a tower is observed to be 60°. At a point, 30 m vertically above the first point of observation, the elevation is found to be 45°. Find: (i) the height of the tower, (ii) its horizontal distance from the points of observation.

Solve for the height and horizontal distance of a tower using trigonometric functions. Angles of elevation are given as 60° and 45°, with a vertical difference of 30 meters. This solution involves key trigonometric concepts suitable for Grades 10-12.

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