1. A circus artist is climbing a 20m 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 (see figure).
1. A circus artist is climbing a 20m 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 (see figure).
Explanation:
In right angle triangle ABC,
AB = 10 m
Therefore, the pole is 10 metres tall.
2. A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle 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.
2. A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle 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.
Explanation:
In right angle triangle ABC,
m
In the same way,
AB = m
Tree height = AB + AC
= =
= = m
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 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 to the ground. What should be the length of the slide in each case?
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 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 to the ground. What should be the length of the slide in each case?
Explanation:
In right angle triangle ABC,
AC = 3 m
In right angle triangle PQR,
PR = m
Hence, the slides are 3 m and m long, respectively.
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 lower is . Find the height of the tower.
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 lower is . Find the height of the tower.
Explanation:
In right angle triangle ABC,
AB = m
= m
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 Find the length of the string, assuming that there is no slack in the string.
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 Find the length of the string, assuming that there is no slack in the string.
Explanation:
In right angle triangle ABC,
AC = m
The string, therefore, has a length of m.
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 to as he walks toward the building. Find the distance he walked towards the building.
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 to as he walks toward the building. Find the distance he walked towards the building.
Explanation:
AB = 30 m and PR = 1.5 m
AC = AB – BC
= AB – PR
= 30 – 1.5
= 28.5 m
In right angle triangle ACQ,
QC = m
In right angle triangle ACP,
PQ =
PQ = = m
Thus, the boy walked toward the direction of the building is m.
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 and respectively. Find the height of the tower.
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 and respectively. Find the height of the tower.
Explanation:
Let the tower's height be m. Afterward, in the right triangle CBP,
……….(i)
In right angle triangle ABP,
BP = 20 m
To put this value in eq. (i), we achieve,
=
m
The tower's elevation is m.
8. A statue, 1.6 m tall, stands on the top of a postal. From a point on the ground, the angle of elevation of the top of the statue is and from the same point the angle of elevation of the top of the pedestal is Find the height of the pedestal.
8. A statue, 1.6 m tall, stands on the top of a postal. From a point on the ground, the angle of elevation of the top of the statue is and from the same point the angle of elevation of the top of the pedestal is Find the height of the pedestal.
Explanation:
Allow the pedestal's height to be m.
BC = m
In right angle triangle ACP,
……….(i)
In right angle triangle BCP,
PC =
[From eq. (i)]
m
Consequently, the pedestal's height is m.
9. The angle of elevation of the top of a building from the foot of the tower is and the angle of elevation of the top of the tower from the foot of the building is If the tower is 50 m high, find the height of the building.
9. The angle of elevation of the top of a building from the foot of the tower is and the angle of elevation of the top of the tower from the foot of the building is If the tower is 50 m high, find the height of the building.
Explanation:
Allow the building's height be m.
BQ = m……….(i)
In right angle triangle ABQ,
BQ = m……….(ii)
From eq. (i) and (ii),
m
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 and respectively. Find the height of the poles and the distances of the point from the poles.
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 and respectively. Find the height of the poles and the distances of the point from the poles.
Explanation:
In right angle triangle PRQ,
H = m……….(i)
In right angle triangle ABR,
[From eq. (i)]
m
H = = m
Also, BR = = 80 – 20 = 60 m
As a result, the poles are each m in height, and the distances between the poles and the point are 20 m and 60 m, respectively.
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 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 (see figure). Find the height of the tower and the width of the canal.
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 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 (see figure). Find the height of the tower and the width of the canal.
Explanation:
In right angle triangle ABC,
AB = m……….(i)
In right angle triangle ABD,
AB = m……….(ii)
From eq. (i) and (ii),
=
3BC = BC + 20
BC = 10 m
From eq. (i), AB = m
As a result, the tower has a height of m and a breadth of 10 m.
12. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is and the angle of depression of its foot is Determine the height of the tower.
12. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is and the angle of depression of its foot is Determine the height of the tower.
Explanation:
In right angle triangle ABD,
BD = 7 m
AE = 7 m
In right angle triangle AEC,
CE = m
CD = CE + ED
= CE + AB
= = m
The tower's height is thus m.
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 two ships.
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 two ships.
Explanation:
In right angle triangle ABQ,
BQ = 75 m……….(i)
In right angle triangle ABP,
[From eq. (i)]
75 + QP =
QP = m
Therefore, the separation between the two ships is m.
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 distant is After some time, the angle of elevation reduces to (see figure). Find the distance travelled by the balloon during the interval.
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 distant is After some time, the angle of elevation reduces to (see figure). Find the distance travelled by the balloon during the interval.
Explanation:
In right angle triangle ABC,
BC = m
In right angle triangle PQC,
= 176.4
BQ = =
= = = m
Hence, the balloon's distance traveled throughout the interval equals m.
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 , 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 Find the time taken by the car to reach the foot of the tower from this point.
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 , 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 Find the time taken by the car to reach the foot of the tower from this point.
Explanation:
In right angle triangle ABP,
BP = AB ……….(i)
In right angle triangle ABQ,
BQ = ……….(ii)
PQ = BP – BQ
PQ = AB
= = = 2BQ [From eq. (ii)]
BQ = PQ
The duration of the car's distance-traveling PQ trip = 6 seconds.
Duration of a distance-traveling vehicle BQ, i.e. PQ = x 6 = 3 seconds.
As a result, it takes the car an additional 3 seconds to arrive at the tower's base.
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.
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.
Explanation:
Let us consider APB =
By this, AQB =
[APB and AQB are complementary]
In right angle triangle ABP,
……….(i)
In right angle triangle ABQ,
=
……….(ii)
Multiplying eq. (i) and eq. (ii),
AB2 = 36
AB = 6 m
Thus, the height of the tower is 6 m.
Proved.
CHAPTER 9 Some Applications Of Trigonometry