A balloon is connected to a meteorological ground station by a cable of length $ 215 \mathrm{~m} $ inclined at $ 60^{\circ} $ to the horizontal. Determine the height of the balloon from the ground. Assume that there is no slack in the cable.
Given:
A balloon is connected to a meteorological ground station by a cable of length \( 215 \mathrm{~m} \) inclined at \( 60^{\circ} \) to the horizontal.
To do:
We have to determine the height of the balloon from the ground.
Solution:
Let $C$ be the meteorological ground station and $AB$ be the height of the balloon from the ground.
From the figure,
$\mathrm{BC}=215 \mathrm{~m}, \angle \mathrm{BCA}=60^{\circ}$
Let the height of the balloon from the ground be $\mathrm{AB}=h \mathrm{~m}$.
We know that,
$\sin \theta=\frac{\text { Opposite }}{\text { Hypotenuse }}$
$=\frac{\text { AB }}{BC}$
$\Rightarrow \sin 60^{\circ}=\frac{h}{215}$
$\Rightarrow \frac{\sqrt3}{2}=\frac{h}{215}$
$\Rightarrow (215)\frac{\sqrt3}{2}=h \mathrm{~m}$
$\Rightarrow h=\frac{215(1.732)}{2} \mathrm{~m}$
$\Rightarrow h=185.975 \mathrm{~m}$
$\Rightarrow h=186 \mathrm{~m}$
Therefore, the height of the balloon from the ground is $186 \mathrm{~m}$.
Related Articles
- The lower window of a house is at a height of \( 2 \mathrm{~m} \) above the ground and its upper window is \( 4 \mathrm{~m} \) vertically above the lower window. At certain instant the angles of elevation of a balloon from these windows are observed to be \( 60^{\circ} \) and \( 30^{\circ} \) respectively. Find the height of the balloon above the ground.
- A kite is flying at a height of $60\ m$ above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is $60^o$. Find the length of the string, assuming that there is no slack in the string.
- A kite is flying at a height of 75 metres from the ground level, attached to a string inclined at \( 60^{\circ} \) to the horizontal. Find the length of the string to the nearest metre.
- A $1.2\ m$ tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2\ m$ from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60^o$. After sometime, the angle of elevation reduces to $30^o$ (see figure). Find the distance travelled by the balloon during the interval."
- From the top of a \( 7 \mathrm{~m} \) high building, the angle of elevation of the top of a cable tower is \( 60^{\circ} \) and the angle of depression of its foot is \( 45^{\circ} . \) Determine the height of the tower.
- A carpenter makes stools for electricians with a square top of side \( 0.5 \mathrm{~m} \) and at a height of \( 1.5 \mathrm{~m} \) above the ground. Also, each leg is inclined at an angle of \( 60^{\circ} \) to the ground. Find the length of each leg and also the lengths of two steps to be put at equal distances.
- A vertically straight tree, \( 15 \mathrm{~m} \) high, is broken by the wind in such a way that its top just touches the ground and makes an angle of \( 60^{\circ} \) with the ground. At what height from the ground did the tree break?
- A tower stands vertically on the ground. From a point on the ground, \( 20 \mathrm{~m} \) away from the foot of the tower, the angle of elevation of the top of the tower is \( 60^{\circ} \). What is the height of the tower?
- A tree is broken at a height of $5\ m$ from the ground and its top touches the ground at a distance of $12\ m$ from the base of the tree. Find the original height of the tree.
- A circus artist is climbing a \( 20 \mathrm{~m} \) long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is \( 30^{\circ} \).
- A ladder is placed along a wall of a house such that its upper end is touching the top of the wall. The foot of the ladder is \( 2 \mathrm{~m} \) away from the wall and the ladder is making an angle of \( 60^{\circ} \) with the level of the ground. Determine the height of the wall.
- A girl of height \( 90 \mathrm{~cm} \) is walking away from the base of a lamp-post at a speed \( 1.2 \mathrm{~m} / \mathrm{sec} \). If the lamp is \( 3.6 \mathrm{~m} \) above the ground, find the length of her shadow after 4 seconds.
- A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of $1.5\ m$, and is inclined at an angle of $30^o$ to the ground, whereas for elder children, she wants to have a steep slide at a height of $3\ m$, and inclined at an angle of $60^o$ to the ground. What should be the length of the slide in each case?
- From a point on the ground the angles of elevation of the bottom and top of a transmission tower fixed at the top of \( 20 \mathrm{~m} \) high building are \( 45^{\circ} \) and \( 60^{\circ} \) respectively. Find the height of the transimission tower.
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies