Question

An observer at a distance of 10 m from tree looks at the top of the tree, the angle of elevation is 60\(^\circ\). To find the height of tree complete the activity. (\(\sqrt{3} = 1.73\)) 

Activity : 
In the figure given above, AB = h = height of tree, BC = 10 m, distance of the observer from the tree. 
Angle of elevation (\(\theta\)) = \(\angle\)BCA = 60\(^\circ\) 
tan \(\theta\) = \(\frac{\boxed{\phantom{AB}}}{BC}\) \(\dots\) (I)
tan 60\(^\circ\) = \(\boxed{\phantom{\sqrt{3}}}\) \(\dots\) (II)
\(\frac{AB}{BC} = \sqrt{3}\) \(\dots\) (From (I) and (II))
AB = BC \(\times\) \(\sqrt{3}\) = 10\(\sqrt{3}\)
AB = 10 \(\times\) 1.73 = \(\boxed{\phantom{17.3}}\)
\(\therefore\) height of the tree is \(\boxed{\phantom{17.3}}\) m.

Hint:

Remember the SOH-CAH-TOA mnemonic for trigonometric ratios. When you have the adjacent side and need to find the opposite side (like in height and distance problems), the Tangent ratio (Opposite/Adjacent) is the one to use.

Answer 1

Step 1: Understanding the Concept: 
This problem uses trigonometry to find the height of an object given the distance from the object and the angle of elevation. The situation forms a right-angled triangle where the height is the opposite side and the distance is the adjacent side to the angle of elevation. 
 

Step 2: Key Formula or Approach: 
In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. 

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

Step 3: Detailed Explanation: 
Here is the completed activity with the blanks filled in. 
In the figure given above, AB = h = height of tree, BC = 10 m, distance of the observer from the tree. 
Angle of elevation (\(\theta\)) = \(\angle\)BCA = 60\(^\circ\) 
In right-angled \(\triangle\)ABC, 
tan \(\theta\) = \(\frac{\boxed{AB}}{BC}\) \(\dots\) (I)
We know that, tan 60\(^\circ\) = \(\boxed{\sqrt{3}}\) \(\dots\) (II)
\(\therefore\) \(\frac{AB}{BC} = \sqrt{3}\) \(\dots\) (From (I) and (II))
AB = BC \(\times\) \(\sqrt{3}\) = 10\(\sqrt{3}\)
AB = 10 \(\times\) 1.73 = \(\boxed{17.3}\)
\(\therefore\) height of the tree is \(\boxed{17.3}\) m.

Step 4: Final Answer: 
The height of the tree is 17.3 m.