NCERT Class X Chapter 9: Some Application Of Trigonometry Exercise 9.1 Question 10

NCERT Class X Chapter 9: Some Applications of Trigonometry

Question:

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.

Given:

  • Width of the road = 80 m
  • Angles of elevation of the tops of the poles from a point between them are \(60^\circ\) and \(30^\circ\)
  • The poles are of equal height

To Find:

  • The height of each pole
  • The distances of the point from each pole

Formula:

\[ \tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}} \]

Solution:

Step 1: Let the height of each pole be \( h \) meters. Let the distances from the point to the two poles be \( x \) meters and \( y \) meters, respectively. The road is 80 m wide, so:

\[ x + y = 80 \]

Step 2: Using the angle of elevation \(60^\circ\) for the pole at distance \(x\):

\[ \tan 60^\circ = \frac{h}{x} \] \[ \sqrt{3} = \frac{h}{x} \] \[ x = \frac{h}{\sqrt{3}} \]

Step 3: Using the angle of elevation \(30^\circ\) for the pole at distance \(y\):

\[ \tan 30^\circ = \frac{h}{y} \] \[ \frac{1}{\sqrt{3}} = \frac{h}{y} \] \[ y = h \sqrt{3} \]

Step 4: Substitute the values of \(x\) and \(y\) into the equation \(x + y = 80\):

\[ \frac{h}{\sqrt{3}} + h\sqrt{3} = 80 \] \[ h \left( \frac{1}{\sqrt{3}} + \sqrt{3} \right) = 80 \] \[ h \left( \frac{1 + 3}{\sqrt{3}} \right) = 80 \] \[ h \left( \frac{4}{\sqrt{3}} \right) = 80 \]

Step 5: Solve for \(h\):

\[ h = \frac{80 \sqrt{3}}{4} \] \[ h = 20\sqrt{3} \text{ m} \]

Step 6: Find the distances from the point to each pole:

\[ x = \frac{h}{\sqrt{3}} = \frac{20\sqrt{3}}{\sqrt{3}} = 20 \text{ m} \] \[ y = h\sqrt{3} = 20\sqrt{3} \times \sqrt{3} = 20 \times 3 = 60 \text{ m} \]

Result:

Height of each pole = \( 20\sqrt{3} \) m \( \approx 34.64 \) m
Distances of the point from the poles are 20 m and 60 m.

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