Question:

1. Draw a right‑angled triangle for each slide:
- The vertical side represents the given height (opposite side).
- The angle between the ground (base) and the slide is the given inclination \(\theta\).
- The slide itself is the hypotenuse of the triangle.

2. Use the sine relation:
$$\sin \theta = \frac{\text{height}}{\text{length of slide}} \quad\Rightarrow\quad \text{length of slide}=\frac{\text{height}}{\sin \theta}$$

3. Case (a): Height = 1.5 m, \(\theta = 30^{\circ}\)
- \(\sin 30^{\circ}=\frac{1}{2}=0.5\)
- $$\text{Length}=\frac{1.5}{0.5}=3\text{ m}$$

4. Case (b): Height = 3 m, \(\theta = 60^{\circ}\)
- \(\sin 60^{\circ}=\frac{\sqrt{3}}{2}\approx 0.866\)
- Exact value:
$$\text{Length}=\frac{3}{\frac{\sqrt{3}}{2}}=\frac{6}{\sqrt{3}}=2\sqrt{3}\text{ m}$$
- Approximate value:
$$\text{Length}\approx \frac{3}{0.866}=3.46\text{ m (to two decimal places)}$$

5. State the answers:
- Slide for children <5 years: 3 m long.
- Slide for elder children: $2\sqrt{3}$ m (≈ 3.46 m) long.