13. As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are and . If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.

Some Applications of Trigonometry | Exercise 9.1

Question 13

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.

Solution

Angle of Depression: The angle between the horizontal line of sight and the line looking down to a point below. Here, both angles (45° and 30°) are measured downward from the lighthouse top A to the ships.

Alternate Interior Angles: A horizontal line AX from A is parallel to the sea level. The depression angle $\angle XAC$ equals the elevation angle $\angle ACB$ (alternate interior angles with BD as transversal). This lets us use the depression angles directly in the right triangles.

Tangent Ratio: $\tan\theta = \dfrac{\text{Opposite (height)}}{\text{Adjacent (distance)}}$ — used to find each ship's distance from the lighthouse base.

Both ships on the same side: The distance between them = farther distance $-$ closer distance = $BD - BC$ .

Step 1 — Set up the problem

We draw a diagram to represent the situation. Let AB be the lighthouse. Its height is 75 m. Let C be the position of the closer ship. Let D be the position of the farther ship. Both ships are on the same side of the lighthouse. The angles of depression are $45^\circ$ and $30^\circ$ . The angle of depression to the closer ship (C) is $45^\circ$ . The angle of depression to the farther ship (D) is $30^\circ$ . Let AX be the horizontal line from A. The angle $\angle XAC = 45^\circ$ . The angle $\angle XAD = 30^\circ$ . The alternate interior angles are $\angle ACB = 45^\circ$ and $\angle ADB = 30^\circ$ .

Diagram 1

Step 2 — Calculate distance to the closer ship

Let's consider the right triangle $\triangle ABC$ . The height of the lighthouse AB is 75 m. The angle $\angle ACB$ is $45^\circ$ . We use the tangent ratio. $\tan(\angle ACB) = \frac{AB}{BC}$ $\tan(45^\circ) = \frac{75}{BC}$ $1 = \frac{75}{BC}$ $BC = 75 \times 1$

$\boxed{BC = 75 \text{ m}}$ So, the closer ship is 75 m from the lighthouse.

Step 3 — Calculate distance to the farther ship

Now, let's consider the right triangle $\triangle ABD$ . The height of the lighthouse AB is 75 m. The angle $\angle ADB$ is $30^\circ$ . We use the tangent ratio again. $\tan(\angle ADB) = \frac{AB}{BD}$ $\tan(30^\circ) = \frac{75}{BD}$ $\frac{1}{\sqrt{3}} = \frac{75}{BD}$ $BD = 75 \times \sqrt{3}$

$\boxed{BD = 75\sqrt{3} \text{ m}}$ The farther ship is $75\sqrt{3}$ m from the lighthouse.

Step 4 — Find the distance between the ships

The distance between the two ships is CD. We can find CD by subtracting BC from BD. $CD = BD - BC$ $CD = 75\sqrt{3} - 75$ $CD = 75(\sqrt{3} - 1)$ We know that $\sqrt{3} \approx 1.732$ . $CD = 75(1.732 - 1)$ $CD = 75(0.732)$ $CD = 54.9$

Answer

The distance between the two ships is 54.9 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.