HEIGHT AND DISTANCE

Height and Distance

Type -1

1) What is the ratio between the height of a vertical pole and  length of its shadow, when the elevation of the sun is 

A) 30°.    1:√3

B) 45°.      1:1

C) 60°.       √3:1

2) The length of the shadow of x metres high vertical tower is x√3. What is the elevation of the sun ?   30°

3)  What is the angle of the elevation of the sun, when the length of shadow of a vertical Pole is equal to its height ?   45°

4) The height of a tree is √3 times, the length of its Shadow. Find the angle of the elevation the sun.   60

5) The angle of the elevation of the top of a tower from a point on the ground and at a distance of 160 m from its foot, is found to be 60°. Find the height of the tower.   277.12m

6) A kite is attached to a string. Find the length of the string, when the height of the kite 60°m and the string makes an angle of 30° with the ground.         120m

7) A boy, 1.6 m tall, is 20m away from a tower and observes the angle of the elevation of the top of the tower to be 

A) 45°

B) 60°

 Find the height of the tower in each case.           21.6m, 36.24m

8) A vertical flagstaff stands on a horizontal plane. From a point 80 m from its foot, the angle of elevation of its top  is found to be 30°. Find the height of the flagstaff.        46.19m

9) The upper part of a tree, broken over by the wind, makes on angle of 45° with ground; and the distance from the root to the point where the top touches the ground, is 15m. What was the height of the tree before it was broken.             36.21m

10) The angle of Bangalore elevation of the top of an unfinished tower at a point distance 80m from its base is 30°. How much higher must the Tower be raised so that its angle of elevation at the same point may be 60°.       92.37m

11) The angle of elevation of the top of a tower, which is incomplete 45° from a point on the level ground and at a distance of 150 m from the base of the tower. How much higher should it be raised so that the elevation of the top of the tower may become 60° at the same point?  109.8m

12) At a  particular time, when the sun's altitude is 30°, the length of the shadow of a vertical Tower is 45°m. Calculate

A)  the height of the tower.

B) the length of the shadow of the same Tower, when the sun's altitude is 

A) 45°.        25.98m

A)  60°.        25.98m 

Type -2

1) The length of the shadow of a vertical tower on level ground increases by 10m, when the altitude of the sun changes from 45° to 30°. Calculate the height of the tower.         13.66m

2) An observer on the top of a cliff,  200 m above the sea level, observes the angle of a depression of the two ships to be 45° and 30° respectively. Find the distance between the ships, if the ships are

A)  on the same side of the cliff.

B) on the opposite sides of the cliff.      146.4m, 546.4m

3) A man on the top of a vertical observation Tower of a car moving at uniform speed coming directly towards it, if it takes 12 minutes for the angle of depression to change from 30° to 45°. How soon after this will be the car reach the observation Tower ?    16.39 minutes

4) Find the height of a tree when it is found that walking away from it 20 m, in a horizontal line through its base, the elevation of its top changes from 60° to 30°.     17.32m

5) A person standing on the bank of a river observes the angle of elevation of the top of a tree, on a opposite bank, to be 60°. When he retires 30m from the bank, he finds the angle of the elevation to be 30°. Find the height of the tree and the breadth of the river.     25.98m, 15m

6) Find the height of a building, when it is found that on walking towards it 40 m in a horizontal line through its base the angular elevation of its top changes from 30° to 45°.            54.64m

7) The shadow of a tower standing on a level ground is found to be 40m longer, when Sun's altitude decreases from 45°  to 30°. Find the height of the tower.       54.64m

8) Two pillars of equal heights stands on either side of a roadway, which is 150m wide. At a point in the roadway between the pillars the elevations of the tops of the pillars are 60° and 30°, find the height of the pillars and the position of the point.    64.95m, 37.5m

9) The angle of the elevation of the tower of the top of a tower is observed to be 60°. At a point 30m vertically above the first point of the observation, the elevation is found to be 45°. Find

A) the height of the tower.   70.98m

B) its horizontal distance from the points from the points of observation.         40.98m

10) From the top of a cliff, 60m high the angles of depression of the top and bottom of a tower are observed to be 30° and 60°. Find the height of the tower.      40m

11) The angle of elevation of the top P of a vertical Tower PQ from a point X is 60°, at a point Y, 40 m vertically above X, the angle of elevation is 45°.

A) Find the height of the tower PQ.

B) Find the distance XQ.  95, 55

(Give your answers to the nearest metre)

12) A man on a cliff observes a boat at an angle of depression 30°  which is sailing towards the shore to the pointing immediately beneath him.  3 minutes later the angle of depression of the boat is found to be 60°  assuming that the boat sail at a uniform speed, determine:

A) how much more time it will take to reach the shore.   1.5min

B)  the speed of the boat in metres per second. If the height of the cliff is 500m.          3.21 m/s

13) A man in a Boat rowing away from a Lighthouse 150m high, takes 2 minutes to change the angle of elevation of the top of the Lighthouse from 60°  to 45°.  Find the speed of the boat.   0.53 m/sec

14) A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle evelvation to be 30°. Find

A) the height of the tree, correct to 2 decimal places.     34.64m

B) the width of the River.     20m

15) The horizontal distance between the two Towers is 75m and the angular depression of the top of the first tower as seen from the top of the second, which is 160m high, is 45°.  Find the height of the first tower.         85m

16) The length of the shadow of a tower standing on a level plane is found to be 2y metre longer when the sun's altitude is 30°  than when it was 45°. Prove that the height of the tower is y(√3+1) m.

17) An aeroplane flying horizontally 1 km above the ground is observed at an elevation at 60°. After 10 seconds its elevation is observed to the 30°, find the uniform speed of the Aeroplane in km per hour.     415.67 km/hr