NCERT Solution for Class 10 Mathematics Chapter 9 Some Applications of Trigonometry
Chapter Name | NCERT Solution for Class 10 Maths Chapter 9 Some Applications of Trigonometry |
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Short Revision for Some Applications of Trigonometry
- The height of an object and the distance between two distinct objects can be determined by using the trigonometric ratios.
- The line of sight is the line drawn from the eye of an observer to the point in the object.
- The angle of elevation is the angle formed by the line of sight with the horizontal when the object is above the horizontal level.
- The angle of depression is the angle formed by the line of sight with the horizontal when the object is below the horizontal level.
NCERT Exercise Solutions
Exercise 9.1
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°. (see fig. 9.11)
Length of the rope AC = 20 m
In △ABC, ∠ACB = 30°
∴ sin 30° = AB/AC ⇒ 1/2 = x/20 ⇒ x = 10 m
∴ Height of the pole = 10 m.
2. 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.
Solution
Let the height of tree before storm be AB. Due to storm it breaks from C such that its top A touches the ground at D and makes an angle of 30° . 
3. 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 3m, and inclined at an angle of 60° to the ground. What should be the length of the slide in each case?
4. 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.
Solution
Let AB be the tower and C be a point on the ground such that BC = 30 m. Also let AB = x m.
The angle of elevation of the top A from C is 30° .
So, in right - angled △ABC, 
5. 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.
6. 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.
Solution
Let a 1.5 m tall boy DP finds the angle of elevation to the top A of the building A.
Let DCB be the horizontal eye sight. When that boy walks towards the building finds increased angle of 60° at C. 
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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.
Solution
In the figure, AB represents a 7 m high building and PQ represents a cable tower. Angle of elevation to the top P from A is 60° and angle of depression to the bottom Q from same point A is 45°.
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° 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
In the figure, AB represents a 75 m high lighthouse, C and D are the positions of two ships where angles of depression from top of the tower are 45° and 30° respectively.
Let BC = x m and BD = y m.
We have to find the distance between the to ships,
i.e., CD = BD - BC = y - x 
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°. After some time, the angle of elevation reduces to 30° (see Fig. 9.13). Find the distance travelled by the balloon during the interval.
Solution
When the horizontally moving balloon is at P, a girl AB (say) finds ∠PAC is 60° and at the position R, ∠RAD is 30° .
We have to find the distance travelled in horizontal line, i.e., PR of CD.
Consider PC = PQ - CQ = (88.2 - 1.2)m = 87 m = RD 
15. 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.
Solution
Let AB be the tower on the top of which a man is standing and finds angle of depression to the position D of the running car as 30° .
After 6 seconds, angle is found 60° at C.
Let AB = h m, DC = x m and BC = y m. 
16. The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m.
Solution
In the drawn figure, AB represents a tower, C and D are two points distant 4 m and 9 m away from the base B.
Let angle of elevation to the top A from C is α and from D is 90° - α . 