Question

A tree casts a shadow 9.3m long when the angle of the sun is ${{43}^{\circ }}$ . How tall is the tree?

Answer 1

Hint: We are given that the tree casts a shadow 9.3 m long, when the angle of sun is ${{43}^{\circ }}$ , we are asked to find the length of tree, to do this, we will learn what does angle of the sun mean, then we sketch our problem, we will learn about trigonometric ratio, we will use $\tan \theta ,\tan \theta =\dfrac{\text{Perpendicular}}{\text{Base}}$ using this, we will find the value of tree.

Complete step-by-step solution:
We are given that the tree casts a shadow 9.3 m long when the angle of the sun is ${{43}^{\circ }}$ , we have to find the length of the tree.
To do so, we will first when we have this type of problem, we see two types of angle, angle of elevation and other is angle of depression.
In our case, we are given that sun is making an angle ${{43}^{\circ }}$so it is angle of elevation from a point to the highest power, so our problem will be as –

We can see that we have a right angle triangle with the base as 9.3 m and the height of the triangle is nothing but the height of the tree.
So, let us consider the height of the tree to be ‘x’ m, so in our case we can see we have a perpendicular as ‘x’ and base is 9.3 m.
We will find the trigonometric ratio which will be suitable for our needs.
We have base and we need perpendicular.
We know $\tan \theta =\dfrac{\text{Perpendicular}}{\text{Base}}$ .
So, we will use $\tan \theta $ , to solve our problem as our angle is ${{43}^{\circ }}$ , so –
$\tan \theta =\dfrac{\text{Perpendicular}}{\text{Base}}$ .
$\theta ={{43}^{\circ }}$ , Perpendicular = x and Base = 9.3 m.
So,
$\tan \left( {{43}^{\circ }} \right)=\dfrac{x}{9.3}$ .
By simplifying, we get –
$x=\tan \left( {{43}^{\circ }} \right)\times 9.3$
Now, as $\tan \left( {{43}^{\circ }} \right)=0.932$ .
So, we get –
$x=0.932\times 9.3$
So, $x=8.67$ .
Hence, we get a height of 8.67 m.

Note: To find the suitable trigonometric ratios we use the formula $\dfrac{\text{given}}{\text{Base}}$ , we write what is given and what is to be find then we match those with our ratio
$\begin{align}
  & \sin \theta =\dfrac{\text{Perpendicular}}{\text{Hypotenuse}} \\
 & \cos \theta =\dfrac{\text{Base}}{\text{Hypotenuse}} \\
 & \tan \theta =\dfrac{\text{Perpendicular}}{\text{Base}} \\
\end{align}$
As the other ratio are opposite of these, so it is easy to find which ratio we can use to find our required value,
Also the answer should be up to 2 decimal expansions for the better solution.