Question

The shadow of a tree when the angle of elevation of the sun at ${45^ \circ }$ is found to be 20 m longer than when it is ${60^ \circ }$. Find the height of a tree. $\left( {\sqrt 3 = 1.73} \right)$
A) 50 m
B) 27.39 m
C) 47.39 m
D) 40 m

Answer 1

Hint:
Draw a diagram representing the given scenario to understand the problem better. Formulate the equations using properties of the trigonometric identity, such as $\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$ for a right angled triangle. Solve the formed equations for two triangles to find the height of the tree.

Complete step by step solution:
It is given in the question that the length of the shadow when the angle of elevation of the sun is ${45^ \circ }$, is 20 m longer than the length of shadow when the angle of elevation of the sun is ${60^ \circ }$.
The above situation can be represented by a diagram as follows.

Here $TA$ represents the length of the tree, $AB$ represents the length of the shadow when the angle of elevation of the sun is ${60^ \circ }$, and $AC$ is the length of the shadow when the angle of the elevation of the sun is ${45^ \circ }$.
Let us assume the length $AB$be represented by $x$ and the length $TA$be represented by $h$.
In the triangle $TAB$,
Using the trigonometric ratio $\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$,
We can say that $\tan {60^ \circ } = \dfrac{{TA}}{{AB}}$
Upon simplification, we get
$
  \sqrt 3 = \dfrac{h}{x} \\
  x = \dfrac{h}{{\sqrt 3 }} \\
$
Similarly for the triangle $TAC$, we can say
$\tan {45^ \circ } = \dfrac{{TA}}{{AC}}$
Upon simplification,
$
  1 = \dfrac{h}{{x + 20}} \\
  h = x + 20 \\
 $
Substituting the value $x = \dfrac{h}{{\sqrt 3 }}$in the equation $h = x + 20$, we get
$h = \dfrac{h}{{\sqrt 3 }} + 20$
We can solve the equation $h = \dfrac{h}{{\sqrt 3 }} + 20$ to find the value of $h$
$
  h\sqrt 3 = h + 20\sqrt 3 \\
  h\left( {\sqrt 3 - 1} \right) = 20\sqrt 3 \\
  h = \dfrac{{20\sqrt 3 }}{{\sqrt 3 - 1}} \\
  h = \dfrac{{20\left( {1.73} \right)}}{{1.73 - 1}} \\
  h \approx 47.397 \\
 $

Thus the height of the tree is $47.39$cm.
Thus the option C is the correct answer.

Note:
In a right angled triangle, we can say $\tan \theta = \dfrac{{{\text{Perpendicular}}}}{{{\text{Base}}}}$, where $\theta $ is the angle between the base and the hypotenuse. The question can be better understood with the help of a diagram representing the scenario.