14. A tall girl spots a balloon moving with the wind in a horizontal line at a height of from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is . After some time, the angle of elevation reduces to (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

Some Applications of Trigonometry | Exercise 9.1

Question 14

A $1.2\text{ m}$ tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2\text{ m}$ from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60^\circ$ . After some time, the angle of elevation reduces to $30^\circ$ (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

Solution

Understand the Question

The balloon moves horizontally at a constant height of $88.2\text{ m}$ from the ground.
Since the girl is $1.2\text{ m}$ tall, the effective vertical height above her eye level is $88.2 - 1.2 = 87\text{ m}$ .
As the balloon moves away horizontally, the angle of elevation decreases from $60^\circ$ to $30^\circ$ .
We use trigonometry in both right-angled triangles to find the initial and final horizontal distances from the girl, then subtract them to find the distance travelled by the balloon.

Step 1 · Determine the Effective Height

Subtract the girl's height from the balloon's height to find the height above her eye level: Diagram 1

\begin{aligned} \text{Effective height } (h) &= 88.2 - 1.2 \\ &= 87\text{ m} \end{aligned}

Step 2 · Calculate the Initial Horizontal Distance

Let $d_1$ be the initial horizontal distance when the angle of elevation is $60^\circ$ .

\begin{aligned} \tan 60^\circ &= \dfrac{h}{d_1} \\[0.6em] d_1 &= \dfrac{h}{\tan 60^\circ} \\[0.6em] &= \dfrac{87}{\sqrt{3}} \end{aligned}

Rationalising the denominator:

\begin{aligned} d_1 &= \dfrac{87\sqrt{3}}{3} \\[0.6em] &= 29\sqrt{3}\text{ m} \end{aligned}

Step 3 · Calculate the Final Horizontal Distance

Let $d_2$ be the final horizontal distance when the angle of elevation is $30^\circ$ .

\begin{aligned} \tan 30^\circ &= \dfrac{h}{d_2} \\[0.6em] d_2 &= \dfrac{h}{\tan 30^\circ} \\[0.6em] &= \dfrac{87}{\dfrac{1}{\sqrt{3}}} \\[1.1em] &= 87\sqrt{3}\text{ m} \end{aligned}

Step 4 · Find the Distance Travelled by the Balloon

The distance travelled is the difference between the final and initial horizontal distances:

\begin{aligned} \text{Distance travelled} &= d_2 - d_1 \\[0.6em] &= 87\sqrt{3} - 29\sqrt{3} \\[0.6em] &= (87 - 29)\sqrt{3} \\[0.6em] &= 58\sqrt{3}\text{ m} \end{aligned}

Using $\sqrt{3} \approx 1.732$ :

\begin{aligned} \text{Distance} &= 58 \times 1.732 \\[0.6em] &= 100.456\text{ m} \approx 100.46\text{ m} \end{aligned}

Answer

$58\sqrt{3}\text{ m}$

Common Mistakes

Ignoring the Observer's Height: Using the total height $88.2\text{ m}$ directly in the trigonometric ratios instead of the effective height $88.2 - 1.2 = 87\text{ m}$ .
Swapping the Angles: Assigning $30^\circ$ to the nearer position and $60^\circ$ to the farther position; as an object moves farther away horizontally, its angle of elevation decreases.

More questions in Exercise 9.1

A circus artist is climbing a 20 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 $30^\circ$ (see Fig. 9.11).

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle $30^\circ$ with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

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 $30^\circ$ 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 $60^\circ$ to the ground. What should be the length of the slide in each case?

The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is $30^\circ$ . Find the height of the tower.

A kite is flying at a height of $60\text{ m}$ 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 $60^\circ$ . Find the length of the string, assuming that there is no slack in the string.

A $1.5\text{ m}$ tall boy is standing at some distance from a $30\text{ m}$ tall building. The angle of elevation from his eyes to the top of the building increases from $30^\circ$ to $60^\circ$ as he walks towards the building. Find the distance he walked towards the building.

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are $45^\circ$ and $60^\circ$ respectively. Find the height of the tower.

A statue, $1.6\text{ m}$ tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is $60^\circ$ and from the same point the angle of elevation of the top of the pedestal is $45^\circ$ . Find the height of the pedestal.

The angle of elevation of the top of a building from the foot of the tower is $30^\circ$ and the angle of elevation of the top of the tower from the foot of the building is $60^\circ$ . If the tower is 50 m high, find the height of the building.

Q10

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are $60^\circ$ and $30^\circ$ , respectively. Find the height of the poles and the distances of the point from the poles.

Q11

A TV tower stands vertically on a bank of a canal. From a point on the other bank directly opposite the tower, the angle of elevation of the top of the tower is $60^\circ$ . From another point 20 m away from this point on the line joing this point to the foot of the tower, the angle of elevation of the top of the tower is $30^\circ$ (see Fig. 9.12). Find the height of the tower and the width of the canal.

Q12

From the top of a $7\text{ m}$ 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.

Q13

As observed from the top of a $75\text{ m}$ high lighthouse from the sea-level, the angles of depression of two ships are $30^\circ$ and $45^\circ$ . If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Q14

A $1.2\text{ m}$ tall girl spots a balloon moving with the wind in a horizontal line at a height of $88.2\text{ m}$ from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is $60^\circ$ . After some time, the angle of elevation reduces to $30^\circ$ (see Fig. 9.13). Find the distance travelled by the balloon during the interval.

Q15

A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of $30^\circ$ , which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be $60^\circ$ . Find the time taken by the car to reach the foot of the tower from this point.