14. 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 . 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 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 $60^\circ$ . 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

Setup: The balloon moves horizontally at a fixed height. As it moves away, the angle of elevation decreases from 60° to 30°. The girl's eyes are at 1.2 m, so the effective height = balloon height − eye level.

Angle of Elevation: The angle between the horizontal and the line of sight looking up to the balloon.

Tangent Ratio: $\tan\theta = \dfrac{\text{Height (opposite)}}{\text{Horizontal distance (adjacent)}}$ — rearranged as $d = \dfrac{h}{\tan\theta}$ to find the horizontal distance.

Why $d_2 > d_1$ : As the balloon moves away, the angle decreases (30° < 60°), so the horizontal distance increases. The distance traveled = $d_2 - d_1$ .

Let's use trigonometry to find the horizontal distance the balloon traveled.

Step 1 — Determine the effective height

The girl's height is 1.2 m. The balloon's height from the ground is 88.2 m. We subtract the girl's height from the balloon's height. This gives us the height from the girl's eye level.

$= 88.2 - 1.2$

$\boxed{87 \text{ m}}$

Diagram 1

Step 2 — Calculate the initial horizontal distance

Let $h$ be the effective height, which is 87 m. Let $d_1$ be the initial horizontal distance. The initial angle of elevation is 60°. We use the tangent function for the right triangle formed.

$\tan(60^\circ) = \frac{h}{d_1}$

$d_1 = \frac{h}{\tan(60^\circ)}$

$d_1 = \frac{87}{\sqrt{3}}$

Rationalizing: Multiply by $\frac{\sqrt{3}}{\sqrt{3}}$ to clear the surd from the denominator.

$d_1 = \frac{87\sqrt{3}}{3}$

$\boxed{29\sqrt{3} \text{ m}}$

Step 3 — Calculate the final horizontal distance

Let $d_2$ be the final horizontal distance. The final angle of elevation is 30°. We use the tangent function again for the new right triangle.

$\tan(30^\circ) = \frac{h}{d_2}$

$d_2 = \frac{h}{\tan(30^\circ)}$

$d_2 = \frac{87}{1/\sqrt{3}}$

$d_2 = 87\sqrt{3}$

$\boxed{87\sqrt{3} \text{ m}}$

Step 4 — Find the distance traveled by the balloon

The distance traveled is the difference between the final and initial horizontal distances. We subtract $d_1$ from $d_2$ .

$= d_2 - d_1$

$= 87\sqrt{3} - 29\sqrt{3}$

$= (87 - 29)\sqrt{3}$

$= 58\sqrt{3}$

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

$= 58 \times 1.732$

$= 100.456$

Answer

The distance traveled by the balloon is $\boxed{58\sqrt{3} \text{ m}}$ or approximately $\boxed{100.46 \text{ m}}$ .

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 3m, 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 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 m tall boy is standing at some distance from a 30 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 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 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 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 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 $60^\circ$ . 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.