CBSE Class 10 Mathematics Some Applications of Trigonometry Worksheet Set 05

Objective Type Questions: 

Question. The length of the ladder making an angle of 45° with a wall and whose foot is 7 m away from the wall is
(a) \( \frac{7\sqrt{2}}{2} \) m
(b) \( 7\sqrt{2} \) m
(c) \( 14\sqrt{2} \) m
(d) 14 m
Answer: (b) \( 7\sqrt{2} \) m

 

Question. A tower casts a shadow 90 m long and at the same time another tower casts a shadow of 120 m on the ground. If the height of the second tower is 80 m, then the height of the first tower is
(a) 60 m
(b) 55 m
(c) 50 m
(d) 40 m
Answer: (a) 60 m

 

Question. If the angles of elevation of a tower from two points distant \( a \) and \( b \) where \( a > b \) from its foot and in the same straight line from it are 30° and 60°, then the height of the tower is 
(a) \( \sqrt{a + b} \)
(b) \( \sqrt{ab} \)
(c) \( \sqrt{a - b} \)
(d) \( \sqrt{\frac{a}{b}} \)
Answer: (b) \( \sqrt{ab} \)

 

Question. The angles of elevation of two cars, from the car to the top of a building are 45° and 30°. If the cars are on the same side of the building and are 50 m apart, what is the height of the building?
(a) \( 25(\sqrt{3} + 1) \) m
(b) \( 25(\sqrt{3} - 1) \) m
(c) \( 50(\sqrt{3} + 1) \) m
(d) \( 50(\sqrt{3} - 1) \) m
Answer: (a) \( 25(\sqrt{3} + 1) \) m

 

Very Short Answer Questions: 

 

Question. The ratio of the length of a tree and its shadow is \( 1 : \frac{1}{\sqrt{3}} \). What is the Sun's angle of elevation?
Answer: 60°

 

Question. If two towers of height \( h_1 \) and \( h_2 \) subtend angle of 60° and 30° respectively at the mid-point of the line joining their feet, then find \( h_1 : h_2 \). 
Answer: 3 : 1

 

Question. The height of the tower is 100 m. When the angle of elevation of Sun is 30°, then what is the length of shadow of the tower?
Answer: \( 100\sqrt{3} \) m

 

Question. The tops of two poles of height 16 m and 10 m are connected by a wire of length \( l \) metres. If the wire makes an angle of 30° with the horizontal, then find \( l \).
Answer: 12 m

 

Question. An observer, 1.5 m tall, is 28.5 m away from a 30 m high tower. Determine the angle of elevation of the top of the tower from the eye of the observer. 
Answer: 45°

 

Question. The angle of elevation of the top of a tower from a point \( P \) on the ground is \( \alpha \). After walking a distance \( d \) meter towards the foot of the tower, angle of elevation is found to be \( \beta \). Which angle of elevation is greater?
Answer: \( \beta \) is greater

 

Short Answer Questions-I: 

 

Question. A tree is broken by wind. The top struck the ground at an angle of 30° and at a distance of 30 m from the root. Find the height of the whole tree.
Answer: \( 30\sqrt{3} \) m

 

Question. The angle of elevation of a ladder leaning against a wall is 60° and the foot of the ladder is 9.5 m away from the wall. Find the length of the ladder.
Answer: 19 m

 

Short Answer Questions-II: 

 

Question. From a window 15 m high above the ground in a street, the angles of elevation and depression of the top and the foot of another house on the opposite side of the street are 30° and 45°, respectively. Show that the height of the opposite house is 23.66 m. 
Answer: 23.66 m

 

Question. The length of a string between a kite and a point on the ground is 90 metres. If the string makes an angle \( \theta \) with the ground level such that \( \tan \theta = \frac{15}{8} \), how high is the kite? Assume that there is no slack in the string.
Answer: 79.41 m

 

Question. The angle of elevation of the top of a vertical tower from a point on the ground is 60°. From another point 10 m vertically above the first, its angle of elevation is 45°. Find the height of the tower. 
Answer: 23.66 m

 

Question. The angles of depression of the top and bottom of a 50 m high building from the top of a tower are 45° and 60° respectively. Find the height of the tower and the horizontal distance between the tower and the building. (Use \( \sqrt{3} = 1.73 \)).
Answer: 118.25 m, 68.25 m

 

Question. From the top of a 120 m high tower, a man observes two cars on the opposite sides of the tower and in straight line with the base of tower with angles of depression as 60° and 45°. Find the distance between the two cars. (Take \( \sqrt{3} = 1.732 \)) 
Answer: 189.28 m

 

Question. The angles of depression of two ships from an aeroplane flying at the height of 7500 m are 30° and 45°. If both the ships are in the same line and on the same side of the aeroplane such that one ship is exactly behind the other, find the distance between the ships. [Use \( \sqrt{3} = 1.73 \)] 
Answer: 5475 m

 

Question. The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of depression from the top of tower to the foot of hill is 30°. If tower is 50 metre high, find the height of the hill. 
Answer: height of hill = 150 m

 

Question. A man in a boat rowing away from a light house 100 m high takes 2 minutes to change the angle of elevation of the top of the light house from 60° to 30°. Find the speed of the boat in metres per minute. [Use \( \sqrt{3} = 1.732 \)]
Answer: speed of boat = 57.73 m/min

 

Question. The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. Find the distance between the two towers and also the height of the other tower. 
Answer: \( 10\sqrt{3} \) m, 10 m

 

Long Answer Questions: 

 

Question. Amit, standing on a horizontal plane, finds a bird flying at a distance of 200 m from him at an elevation of 30°. Deepak standing on the roof of a 50 m high building, finds the angle of elevation of the same bird to be 45°. Amit and Deepak are on opposite sides of the bird. Find the distance of the bird from Deepak. 
Answer: \( 50\sqrt{2} \) m

 

Question. A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression of the boat changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/min. 
Answer: Speed of boat is \( (75 - 25\sqrt{3}) \) m/min or 31.7 m/min

 

Question. There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole. 
Answer: width of river \( = 20\sqrt{3} \) m and height of pole = 40 m

 

Question. The angles of depression of the top and bottom of a 8 m tall building from the top of a tower are 30° and 45° respectively. Find the height of the tower and the distance between the tower and the building. 
Answer: height of tower \( = 12 + 4\sqrt{3} \) m = distance between tower and building

 

Question. As observed from the top of a lighthouse, 75 m high 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: \( 75(\sqrt{3} - 1) \) m

 

Question. The lower window of a house is at a height of 2 m above the ground and its upper window is 4 m vertically above the lower window. At certain instant, the angles of elevation of a balloon from these windows are observed to be 60° and 30°, respectively. Find the height of the balloon above the ground. 
Answer: 8 m

 

Question. The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of elevation of the top of the tower from the foot of the hill is 30°. If the tower is 50 m high, what is the height of the hill? 
Answer: 150 m

 

Question. The angle of elevation of a cloud from a point 60 m above a lake is 30° and the angle of depression of the reflection of the cloud in the lake is 60°. Find the height of the cloud from the surface of the lake. 
Answer: 120 m

 

Question. A bird is sitting on the top of a tree, which is 80 m high. The angle of elevation of the bird, from a point on the ground is 45°. The bird flies away from the point of observation horizontally and remains at a constant height. After 2 seconds, the angle of elevation of the bird from the point of observation becomes 30°. Find the speed of flying of the bird. 
Answer: 29.28 m/s

 

Question. The angle of elevation of the top \( Q \) of a vertical tower \( PQ \) from a point \( X \) on the ground is 60°. From a point \( Y \), 40 m vertically above \( X \), the angle of elevation of the top \( Q \) of tower is 45°. Find the height of the tower \( PQ \) and the distance \( PX \). (Use \( \sqrt{3} = 1.73 \)) 
Answer: 54.64 m, 94.64 m

 

Question. From a point on the ground, the angle of elevation of the top of a tower is observed to be 60°. From a point 40 m vertically above the first point of observation, the angle of elevation of the top of the tower is 30°. Find the height of the tower and its horizontal distance from the point of observation.
Answer: 60 m; \( 20\sqrt{3} \) m

 

Question. Two points \( A \) and \( B \) are on the same side of a tower and in the same straight line with its base. The angles of depression of these points from the top of the tower are 60° and 45° respectively. If the height of the tower is 15 m, then find the distance between these points. 
Answer: \( 5(3 - \sqrt{3}) \) m