EXERCISE 9.1
Introduction - Some Applications of Trigonometry, Class 10th Maths
Q 1, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Page No 203:
Question 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 °.
Answer:
It can be observed from the figure that AB is the pole.In ΔABC,
Therefore, the height of the pole is 10 m.
Q 2, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 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.Answer:
Let AC was the original tree. Due to storm, it was broken into two parts. The broken part
is making 30° with the ground.In
,
Height of tree =
+ BC
Hence, the height of the tree is
.Q 3, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 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, where as for the elder children she wants to have a steep side 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?Answer:
It can be observed that AC and PR are the slides for younger and elder children respectively.
In ΔABC,
In ΔPQR,
Therefore, the lengths of these slides are 3 m and
.Q 4, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Page No 204:
Question 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.Answer:
Let AB be the tower and the angle of elevation from point C (on ground) is
30°.
In ΔABC,
Therefore, the height of the tower is
.Q 5, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 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.Answer:
Let K be the kite and the string is tied to point P on the ground.
In ΔKLP,
Hence, the length of the string is
.Q 6, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 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.Answer:
Let the boy was standing at point S initially. He walked towards the building and reached at point T.
It can be observed that
PR = PQ − RQ
= (30 − 1.5) m = 28.5 m =
In ΔPAR,
In ΔPRB,
ST = AB
Hence, he walked
towards the building.Q 7, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 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.Answer:
Let BC be the building, AB be the transmission tower, and D be the point on the ground from where the elevation angles are to be measured.
In ΔBCD,
In ΔACD,
Therefore, the height of the transmission tower is
m.Q 8, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 8:
A statue, 1.6 m tall, stands on a top of pedestal, from a point on the ground, the angle of elevation of the top of 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.Answer:
Let AB be the statue, BC be the pedestal, and D be the point on the ground from where the elevation angles are to be measured.
In ΔBCD,
In ΔACD,
Therefore, the height of the pedestal is 0.8
m.Q 9, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 9:
The angle of elevation of the top of a building from the foot of the tower is30° 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.
Answer:
Let AB be the building and CD be the tower.
In ΔCDB,
In ΔABD,
Therefore, the height of the building is
.Q 10, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 10:
Two poles of equal heights are standing opposite each other an 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 30º, respectively. Find the height of poles and the distance of the point from the poles.Answer:
Let AB and CD be the poles and O is the point from where the elevation angles are measured.
In ΔABO,
In ΔCDO,
Since the poles are of equal heights,
CD = AB
DO = BD − BO = (80 − 20) m = 60 m
Therefore, the height of poles is
and the point is 20 m and 60 m far from these poles.Q 11, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 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 joining this point to the foot of the tower, the angle of elevation of the top of the tower is 30°. Find the height of the tower and the width of the canal.
Answer:
In ΔABC,
In ΔABD,
Therefore, the height of the tower is
m and the width of the canal is10 m.
Q 12, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 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.Answer:
Let AB be a building and CD be a cable tower.
In ΔABD,
In ΔACE,
AE = BD = 7 m
Therefore, the height of the cable tower is
.Q 13, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
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° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships.Answer:
Let AB be the lighthouse and the two ships be at point C and D respectively.
In ΔABC,
In ΔABD,
Therefore, the distance between the two ships is
m.Q 14, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Page No 205:
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°. After some time, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during the interval.
Answer:
Let the initial position A of balloon change to B after some time and CD be the girl.
In ΔACE,
In ΔBCG,
Distance travelled by balloon = EG = CG − CE
Q 15, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 15:
A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car as 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.Answer:
Let AB be the tower.
Initial position of the car is C, which changes to D after six seconds.
In ΔADB,
In ΔABC,
Time taken by the car to travel distance DC
= 6 secondsTime taken by the car to travel distance DB
Q 16, Ex 9.1 - Applications of Trigonometry - Chapter 9 - Maths Class 10th - NCERT
Question 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.Answer:
Let AQ be the tower and R, S are the points 4m, 9m away from the base of the tower respectively.
The angles are complementary. Therefore, if one angle is θ, the other will be 90 − θ.
In ΔAQR,
In ΔAQS,
On multiplying equations (i) and (ii), we obtain
However, height cannot be negative.
Therefore, the height of the tower is 6 m.
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