Question

The length of a string between a kite and a point on the roof of a building 10 m high is 180 m. If the string makes an angle $\theta $ with the level ground such that $\tan \theta = \dfrac{4}{3}$, how high is the kite from ground?

Answer 1

Hint: Here, we will first draw the diagram of the problem showing the building and the string attached to a kite. After that by using the trigonometric relations we can find the height of the kite from the ground.

Complete step by step solution:
Given:
The height of the building (from ground level to the roof of the building) is $10\;{\rm{m}}$.
The length of the string attached between the kite and the point on the roof of the building is $180\;{\rm{m}}$.
The angle between the string attached to a kite and the level ground (horizontal) is $\theta $ and also $\tan \theta = \dfrac{4}{3}$.
The following is the schematic diagram of the kite and the building.
.

So, from the above diagram we can find the Sine and Cosine of this angle by using the Trigonometric relations.

In the above diagram, we concluded that,
$\sin \theta = \dfrac{4}{5}$ and $\cos \theta = \dfrac{3}{5}$
Then, the normal (perpendicular) height of the kite from the rooftop of the building is,
 $\begin{array}{c}
 = 180 \times \sin \theta \\
 = 180 \times \dfrac{4}{5}\\
 = 144\;{\rm{m}}
\end{array}$
 The height of the kite from the level ground will be equal to the addition of height of the building (from ground level to the roof of the building) + The normal (perpendicular) height of the kite from the rooftop of the building.
So, the height of the kite from the level ground is,
$\begin{array}{c}
h = 10\;{\rm{m}} + 144\;{\rm{m}}\\
h = 154\;{\rm{m}}
\end{array}$
Therefore, the height of the kite from the level ground is $h = 154\;{\rm{m}}$.


Note: In such types of problems, make sure to remember that the angle $\theta $ shows the inclination of the string of a kite from the ground and it remains constant at any point of the string. The important point for this solution is that the height of the kite is the sum of height of the building and distance of the kite from the terrace of the building.