Question

The length of a string between kite and a point on the ground is \[90{\text{ m}}\]. If the string makes an angle with the level ground and \[\sin \alpha = \dfrac{3}{5}\]. Find the height of the kite. There is no slack in the string.

Answer 1

Hint: Draw a right triangle by using the length of hypotenuse as \[90{\text{ m}}\] and find the angle between string and the ground by using the properties of the triangle. Since the value of \[\sin \alpha = \dfrac{3}{5}\] we will find the other sides of the triangle by comparing it with \[\sin \alpha = \dfrac{{\text{P}}}{{\text{H}}}\] after substituting the values and keeping both the angles we will be able to find the value of the height of the kite.

Complete step by step solution
We will first consider the given data that is length of the string is \[90{\text{ m}}\] and \[\sin \alpha = \dfrac{3}{5}\].
To find the height of the kite from the ground, first find the angle between ground and string.
Draw a right triangle having vertices A, B and C.



In the above triangle,
Length that is \[{\text{AC}} = 90{\text{ m}}\] is given in the question.
Let the angle between string and ground is \[\alpha \], that is \[\angle {\text{ACB}} = \alpha \].
Also, we know that \[\sin \alpha = \dfrac{3}{5}\]
Let AB be the height of the kite from the ground, denote the height by \[h\].
By using the properties of triangles, it is known that \[\sin \alpha = \dfrac{{\text{P}}}{{\text{H}}}\].
Now, on comparing \[\sin \alpha = \dfrac{{\text{P}}}{{\text{H}}}\] with \[\sin \alpha = \dfrac{3}{5}\],
we get,
\[ \Rightarrow \dfrac{{\text{P}}}{{\text{H}}} = \dfrac{{{\text{AB}}}}{{{\text{AC}}}} = \dfrac{3}{5}\]
which further gives us
\[ \Rightarrow \dfrac{{{\text{AB}}}}{{{\text{AC}}}} = \dfrac{3}{5}\].
Next, we will substitute \[h\] for AB and 90 for AC in \[\dfrac{{{\text{AB}}}}{{{\text{AC}}}} = \dfrac{3}{5}\].
Thus, we get,
\[ \Rightarrow \dfrac{h}{{90}} = \dfrac{3}{5}\]
Now, we will perform the cross multiplication to evaluate the value of \[h\].
Thus, we get,
\[
   \Rightarrow 5h = 90 \times 3 \\
   \Rightarrow h = \dfrac{{90 \times 3}}{5} \\
   \Rightarrow h = 18 \times 3 \\
   \Rightarrow h = 54 \\
 \]
Hence, the height of the kite from the ground is \[54{\text{ m}}\].

Note: Do not use the properties of the triangle in the form of cosecant, because we have to compare with the given sine angle. Use the property that \[\sin \alpha = \dfrac{{\text{P}}}{{\text{H}}}\] and compare it with the values given in the question. Making a figure gives us an idea of what is given and what we need to find.