Question

A balloon is connected to a meteorological ground station by a cable of 215 meters at $60{}^\circ $ to the horizontal. Determine the height of the balloon from the ground. Assume that there is no slack in the cable.

Answer 1

Hint: Assume that the height of the tower from the ground is ‘h’. First, draw a rough diagram of the given conditions and then use the formula $\sin \theta =\dfrac{\text{perpendicular}}{hypotenuse}$ in the different right angle triangles and substitute the given values to get the height.

Complete step-by-step answer:
Let us start with the question by drawing a representative diagram of the situation given in the question.

According to the above figure:

We have assumed the height of the balloon from the ground as ‘h’. Therefore, AB = h.

Now, in right angle triangle ADB,

$\angle ADB=60{}^\circ $

We know that, $\sin \theta =\dfrac{\text{perpendicular}}{hypotenuse}$. Therefore,

$\begin{align}

  & \sin 60{}^\circ =\dfrac{AB}{AD} \\

 & \Rightarrow \tan 60{}^\circ =\dfrac{h}{AD} \\

 & \Rightarrow h=AD\sin 60{}^\circ \\

\end{align}$

And we know that the value of $\sin 60{}^\circ $ is equal to $\dfrac{\sqrt{3}}{2}$ .

$\therefore h=AD\dfrac{\sqrt{3}}{2}$

Now as it is given that the length of the string is 215 meters, so the hypotenuse AD is equal
to 215 meters. If we put the value of AD in our equation, we get

 $h=\dfrac{215\sqrt{3}}{2}$

So, the height of the balloon from the ground is $\dfrac{215\sqrt{3}}{2}$ meters.

Note: Do not use any other trigonometric function like tangent or cosine of the given angle because the information which is provided to us is related to the hypotenuse of the triangle, and we have to find the height. So, the length of the base is of no use. Therefore, the formula of the sine of the angle is used. We can use tangent or cosine of the given angles but then the process of finding the height will be lengthy.