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Q13:
As observed from the top of a $75$ m high lighthouse from the sea-level, the angles of depression of two ships are $30°$ and $45°$. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.
Solution :
Given:
- Height of the lighthouse ($AB$) = $75$ m.
- Angle of depression of the first ship ($C$) = $45^\circ$.
- Angle of depression of the second ship ($D$) = $30^\circ$.
- The ships are on the same side of the lighthouse and lie on the same straight line.
To Find:
The distance between the two ships, i.e., the length of segment $CD$.
Step 1: Defining Variables and Geometric Relationships
Let $AB$ be the lighthouse of height $75$ m. Let $C$ and $D$ be the positions of the two ships. Since the angles of depression are $45^\circ$ and $30^\circ$, the angles of elevation from the ships to the top of the lighthouse are also $45^\circ$ and $30^\circ$ respectively [By the property of alternate interior angles].
Let $BC = x$ and $BD = y$. The distance between the ships is $CD = y - x$.
Step 2: Calculating distance $BC$ in $\triangle ABC$
In right-angled triangle $\triangle ABC$:
$\tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC}$
We know $\tan(45^\circ) = 1$ and $AB = 75$ m.
$1 = \frac{75}{x}$
$x = 75$ m
Step 3: Calculating distance $BD$ in $\triangle ABD$
In right-angled triangle $\triangle ABD$:
$\tan(30^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BD}$
We know $\tan(30^\circ) = \frac{1}{\sqrt{3}}$ and $AB = 75$ m.
$\frac{1}{\sqrt{3}} = \frac{75}{y}$
$y = 75\sqrt{3}$ m
Step 4: Finding the distance between the ships ($CD$)
The distance between the two ships is $CD = BD - BC$.
$CD = y - x$
$CD = 75\sqrt{3} - 75$
$CD = 75(\sqrt{3} - 1)$
Using the approximation $\sqrt{3} \approx 1.732$:
$CD = 75(1.732 - 1)$
$CD = 75(0.732)$
$CD = 54.9$ m
Final Answer: The distance between the two ships is $75(\sqrt{3} - 1)$ m or approximately $54.9$ m.
More Questions from Class 10 Mathematics Applications of Trigonometry EXERCISE 9.1
- Q1: 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 $30°$ (see Fig. 9.11).
- 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 $60°$ and $30°$, 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 $60°$. 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 $30°$ (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 $60°$ and the angle of depression of its foot is $45°$. Determine the height of the tower.
- 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 $60°$. After some time, the angle of elevation reduces to $30°$ (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 $30°$, 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 $60°$. Find the time taken by the car to reach the foot of the tower from this point.
- Q2: A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle $30°$ 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.
- Q3: 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 $30°$ 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 $60°$ to the ground. What should be the length of the slide in each case?
- Q4: 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 $30°$. Find the height of the tower.
- Q5: 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 $60°$. Find the length of the string, assuming that there is no slack in the string.
- Q6: 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 $30°$ to $60°$ as he walks towards the building. Find the distance he walked towards the building.
- Q7: 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 $45°$ and $60°$ respectively. Find the height of the tower.
- Q8: 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 $60°$ and from the same point the angle of elevation of the top of the pedestal is $45°$. Find the height of the pedestal.
- Q9: The angle of elevation of the top of a building from the foot of the tower is $30°$ and the angle of elevation of the top of the tower from the foot of the building is $60°$. If the tower is $50$ m high, find the height of the building.
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