Advertisements
Advertisements
Question
Find the height of a tree when it is found that on walking away from it 20 m, in a horizontal line through its base, the elevation of its top changes from 60° to 30°.
Solution
Let AB be the tree and its height be x
DC = 20 m.
Now in right ΔADB,
`tan theta = (AB)/(DB)`
`\implies tan 30^circ = x/(DB)`
`\implies 1/sqrt(3) = x/(DB)`
`\implies DB = sqrt(3)x`. ...(i)
In ΔACB, we have
`tan 60^circ = x/(CB)`
`\implies sqrt(3)/1 = x/(CB)`
∴ `CB = x/sqrt(3) = (sqrt(3)x)/3` ...(ii)
But DB – CB = DC
`\implies sqrt(3)x - (sqrt(3)x)/3 = 20`
`\implies (3sqrt(3)x - sqrt(3)x)/3 = 20`
`\implies (2sqrt(3)x)/3 = 20`
`\implies x = (20 xx 3)/(2sqrt(3))`
= `(10 xx 3 xx sqrt(3))/(sqrt(3) xx sqrt(3))`
= `(30sqrt(3))/3`
∴ `x = 10sqrt(3)`
= 10 × (1.732)
= 17.32 m.
∴ Required height of the tree = 17.32 m
APPEARS IN
RELATED QUESTIONS
Two vertical poles are on either side of a road. A 30 m long ladder is placed between the two poles. When the ladder rests against one pole, it makes angle 32°24′ with the pole and when it is turned to rest against another pole, it makes angle 32°24′ with the road. Calculate the width of the road.
From the top of a cliff, 60 metres 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.
Calculate AB.
In the given figure, from the top of a building AB = 60 m hight, the angle of depression of the top and bottom of a vertical lamp post CD are observed to be 30° and 60° respectively. Find:
- the horizontal distance between AB and CD.
- the height of the lamp post.
Two persons standing on opposite sides of a tower observe the angles of elevation of the top of the tower to be 60° and 50° respectively. Find the distance between them, if the height of the tower is 80m.
A parachutist is descending vertically and makes angles of elevation of 45° and 60° from two observing points 100 m apart to his right. Find the height from which he falls and the distance of the point where he falls on the ground from the nearest observation pcint.
In triangle ABC, AB = 12 cm, LB = 58°, the perpendicular from A to BC meets it at D. The bisector of angle ABC meets AD at E. Calculate:
(i) The length of BD;
(ii) The length of ED.
Give your answers correct to one decimal place.
From a lighthouse, the angles of depression of two ships on opposite sides of the lighthouse were observed to be 30° and 45°. If the height of the lighthouse is 90 metres and the line joining the two ships passes through the foot of the lighthouse, find the distance between the two ships, correct to two decimal places.
A man on the deck of a ship is 10 m above water level. He observes that the angle of elevation of the top of a cliff is 42° and the angle of depression of the base is 20°. Calculate the distance of the cliff from the ship and the height of the cliff.
An aeroplane when 3,000 meters high passes vertically above another aeroplane at an instance when their angles of elevation at the same observation point are 60° and 45° respectively. How many meters higher is the one than the other?
