A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30° to the ground whereas for elder children, she wants to have a steep slide at a height of 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?

Let AC be the slide for children below 5 years and DF be the slide for elder children.

In △ABC,

sin 30° = $\dfrac{\text{Side opposite to angle 30°}}{\text{Hypotenuse}}$

Substituting values we get :

$\Rightarrow \dfrac{1}{2} = \dfrac{AB}{AC} \\[1em] \Rightarrow \dfrac{1}{2} = \dfrac{1.5}{AC} \\[1em] \Rightarrow AC = 1.5 \times 2 = 3 \text{ m}.$

In △DEF,

sin 60° = $\dfrac{\text{Side opposite to angle 60°}}{\text{Hypotenuse}}$

Substituting values we get :

$\Rightarrow \dfrac{\sqrt{3}}{2} = \dfrac{DE}{DF} \\[1em] \Rightarrow \dfrac{\sqrt{3}}{2} = \dfrac{3}{DF} \\[1em] \Rightarrow DF = \dfrac{3 \times 2}{\sqrt{3}} = 2\sqrt{3} \text{ m}.$

Hence, for children below 5 years, length of slide = 3 m and for elder children, length of slide = $2\sqrt{3}$ m.