Question

The string of kites is 100m long and it makes an angle of \[{60^ \circ }\] with the horizontal. Find the height of the kite assuming that there is no slack in string.

Answer 1

Hint:
This problem is related to heights and distances. In this problem we will use trigonometric functions to find the height of the kite. We are given the angle that the string makes with the horizontal and the length of string of kite that will act as a hypotenuse of the triangular figure. Let’s solve it!

Complete step by step solution:
Let’s diagramise the situation first.

Now as shown in the diagram above, suppose the kite is flying at a height of h meters. The length of the kite string is l=100m. And the angle it makes with the horizontal is \[{60^ \circ }\].
Now in order to find the height h=? we have to take help of trigonometric functions.
From figure,
\[
   \Rightarrow \sin {60^ \circ } = \dfrac{{opposite{\text{ }}side}}{{hypotenuse}} \\
   \Rightarrow \dfrac{{height{\text{ }}of{\text{ }}kite}}{{length{\text{ }}of{\text{ }}kite{\text{ }}string}} \\
 \]
\[ \Rightarrow \dfrac{h}{l}\]
Putting the values
\[ \Rightarrow \sin {60^ \circ } = \dfrac{h}{{100}}\]
\[ \Rightarrow \dfrac{{\sqrt 3 }}{2} = \dfrac{h}{{100}}\]
\[ \Rightarrow h = \dfrac{{100\sqrt 3 }}{2}\]
On dividing
\[ \Rightarrow h = 50\sqrt 3 meters\]

This is the height of the kite \[ \Rightarrow h = 50\sqrt 3 meters\]

Note:
Here the angle it makes with horizontal is the clue we get to use the trigonometric ratios. We have used sine function because we are finding the side opposite to a given angle. And that is only obtained in sine function.