Ex.9.1 Q3 Some Applications of Trigonometry Solution - NCERT Maths Class 10

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Question

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 \(1.5\,\rm{m} \) and is inclined at an angle of \(30^\circ\) to the ground, whereas for elder children she wants to have a steep slide at a height of \(3\,\rm{m}\) and inclined at an angle of \(60^\circ\) to the ground. What should be the length of the slide in each case?

Text Solution

What is Known?

(i) For the children below the age of \(5\) years.

Height of the slide \(=\) \(1.5\,\rm{m} \)

Slide’s angle with the ground \(=\) \(30^\circ\)

What is Unknown?

Length of the slide for children below the age of \(5\) years and elder children.

(ii) For elder children.

           Height of the slide \(= 3\,\rm{m}\)

Slide’s angle with the ground \(= \)\(60^\circ\)

Reasoning:

Let us consider the following conventions for the slide installed for children below \(5\) years :

  • The height of the slide as \(AC.\)
  • Distance between the foot of the slide to the point where it touches the ground as \(AB.\)
  • Length of the slide as \(BC.\)

Let us consider the following conventions for the slide installed for elder children:

  • The height of the slide \(PR.\)
  • Distance between the foot of the slide to the point where it touches the ground as \(PQ.\)
  • Length of the slide as \(QR.\)

(i) Trigonometric ratio involving \(AC, BC\) \(\angle B\) and is \(\sin \theta\)

(ii) Trigonometric ratio involving \(PR, QR\) and \(\angle Q\) and is \(\sin \theta\)

Steps:

(i) In \(\Delta ABC\),

\[\begin{align} \sin 30^{\circ} &=\frac{A C}{B C} \\ \frac{1}{2} &=\frac{1.5}{B C} \\ B C &=1.5 \times 2 \\ B C &=3 \mathrm{m}\end{align}\]

(ii) In \(\Delta \rm{PRQ}\),

\[\begin{align}\sin \,Q &= \frac{{PR}}{{QR}}\\\sin \,60^\circ & = \frac{3}{{QR}}\\\frac{{\sqrt 3 }}{2}& = \frac{3}{{QR}}\\QR &= \frac{{3 \times 2}}{{\sqrt 3 }}\\&= \frac{6}{{\sqrt 3 }} \times \frac{{\sqrt 3 }}{{\sqrt 3 }}\\&= \frac{{6\sqrt 3 }}{3}\,\\&= 2\sqrt 3 \end{align}\]

Length of slide for children below \(5\) years \(= 3\rm{m}.\)

Length of slide for elder children \(=\) \(2 \sqrt3\,\rm{m}\)