Students can Download Class 10 Maths Chapter 12 Some Applications of Trigonometry Ex 12.1 Questions and Answers, Notes Pdf, KSEEB Solutions for Class 10 Maths helps you to revise the complete Karnataka State Board Syllabus and to clear all their doubts, score well in final exams.
Karnataka State Syllabus Class 10 Maths Chapter 12 Some Applications of Trigonometry Ex 12.1
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° (see Fig. 12.4)
Answer:
Let the height of the pole is AB = ‘x’ and AC is the length of rope tied from the top of a vertical pole to the ground
∴AC = 20 m
AB = 10m
∴ height of pole is 10 m.
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 the height of the tree be AD = hm
Broken part of the tree be BC = x
height of the tree remaining is AB = Y
AD = 8 \(\sqrt{3}\)m
∴ height of tree is 8 \(\sqrt{3}\) m.
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 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?
Answer:
Let the children below 5 years slide is DE and the elder children slide is AC.
DE = 1.5 × 2
DE = 3 m
∴ Length of slide for children below 5 years is 3 m
∴ Length of slide for elder children is 2\(\sqrt{3}\) m
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
Let the height of the tower be ‘AC’ and distance foot of the tower to a point AB = 30m.
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 AB be the distance between horizontal ground and point C position and AC is the height of string length.
∴ AC = x
∴ Length of string is 40\(\sqrt{3}\)m
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 AB be the height of building and PQ be the initial position of the boy such that ∠APR = 30° and AB = 30m
Now, let the new position of the boy be P Q.
= (28.5 – 9.5)\(\sqrt{3}\) = 19.0\(\sqrt{3}\)
= 19\(\sqrt{3}\)m
Distance walked by the boy is 19\(\sqrt{3}\)m
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 height of tower BC = h m and AB is the height of building AB = 20 m
h = 20(\(\sqrt{3}\) – 1)m
Question 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.
Answer:
Let the height of the Pedestal BC be ‘h’ m
Height of the statue = AB = 1.6m
h = 0.8(\(\sqrt{3}\) – 1)m
∴ height of the Pedestal is 0.8(\(\sqrt{3}\) – 1)m.
Question 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.
Answer:
Let AB be the height of building AB = ‘h’ m and CD is the height of building CD = 50m.
h = 16\(\frac{2}{3}\)m.
Hence height of Building is 16\(\frac{2}{3}\)m.
Question 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.
Answer:
Let AB and PQ be two poles of equal height h metre and let C be any Point between the two poles.
3x = 80 – x
3x + x = 80
4x = 80
x = \(\frac{80}{4}\) = 20
x = 20 m
Put x = 20 in equation (1)
∴ h = \(\sqrt{3}\)x
h = \(\sqrt{3}\) × 20
h = 20\(\sqrt{3}\)m
Hence, the height of the pole is 20\(\sqrt{3}\) m and the distance of the point from first pole is 20 m and that of second pole is = 80 – x = 80 – 20 = 60 m
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° (see Fig. 12.12). Find the height of the tower and the width of the canal.
Answer:
Let height of the tower be ‘h’ metres and width of the canal be ‘x’ metres, so AB = h m and BC = x m
20 + x = 3x
20 = 3x – x
2x = 20
x= \(\frac{20}{2}\) = 10
x = 10 m.
Put x = 10 in equation (1)
h = \(\sqrt{3}\) × 10
h = 1o\(\sqrt{3}\)m.
Hence, height of the tower is 10\(\sqrt{3}\)m and width of the canal is 10 m.
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 the height of tower
AB = (h + 7)m and PQ be the height of building
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 the height of light house be AB = 75m and distance between two ships be DC = x
Hence, the distance between the two ships is 75(\(\sqrt{3}\) – 1)m
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° (see Fig. 12.13), Fitid the distance travelled by the balloon during the interval.
Answer:
Let A and B be two positions of the balloon and G be a point of observation
∴ AC = BD = BQ – DQ = 88.2 m – 1.2 m = 87 m
GD = 87\(\sqrt{3}\)
CD = GD – GC
= 87\(\sqrt{3}\) – 29\(\sqrt{3}\) = 58\(\sqrt{3}\)m
= 58\(\sqrt{3}\)m
Hence, the balloon travels 58\(\sqrt{3}\)m.
Question 15.
A straight highway leads to the foot of a tower. A irian 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.
Answer:
In right ∆ ABP
BP = 3 BQ
BQ + PQ = 3BQ [From Fig BP = BQ + PQ]
PQ = 3BQ – BQ
PQ = 2 BQ
BQ = 1/2 PQ
∴ time taken by car to travel distance PQ = 6 seconds.
∴ time taken by a car to travel distance BQ = 1/2 PQ = 1/2 × 6 = 3 seconds.
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:
