Advertisements
Advertisements
प्रश्न
If the angle of elevation of a cloud from a point h meters above a lake is a*and the angle of depression of its reflection in the lake is |i. Prove that the height of the cloud is `(h (tan β + tan α))/(tan β - tan α)`.
उत्तर
Let LM be the upper surface of the lake and A be a point such that AL = h.
Let C be the position of the cloud and C' be its reflection in the lake.
CM = MC' = x(let)
∠BAC = α and ∠BAC' = β
Now In ΔCBA,
tan α = `"CB"/"AB"`
tan α = `(x - h)/"AB"`
AB = `(x - h)/(tan α)` .....(i)
In ΔC'BA,
tan β = `"CB"/"AB"`
tan β = `(x + h)/"AB"`
AB = `(x + h)/tan β` .....(ii)
From (i) and (ii),
`(x - h)/(tan α) = (x + h)/tan β`
or `( x + h)/(x - h) = (tanβ)/(tan α)`
App. componendo and dividendo,
`( x + h + x - h )/(x + h - x + h) = (tanβ + tan α)/(tanβ - tan α)`
`( 2x)/(2h) = (tanβ + tan α)/(tanβ - tan α)`
`x = (h(tanβ + tan α))/(tanβ - tan α)`
∴ Height of the cloud is `x = (h(tanβ + tan α))/(tanβ - tan α)` ....Hence proved.
APPEARS IN
संबंधित प्रश्न
A man stands 9 m away from a flag-pole. He observes that angle of elevation of the top of the pole is 28° and the angle of depression of the bottom of the pole is 13°. Calculate the height of the pole.
Find the length of the shadow cast by a tree 60 m high when the sun's altitude is `30^circ`.
The height of an observation tower is 180m above sea level. A ship coming towards the tower is observed at an angle of depression of 30°. Calculate the distance of the boat from the foot of the observation tower.
An aeroplane when flying at a height of 4km from the ground passes vertically above another aeroplane at an instant when the angles of the elevation of the two planes from the same point on the ground are 60° and 45° respectively. Find the vertical distance between the aeroplanes at that instant.
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.
The angle of elevation of an aeroplane from a point on the ground is 45°. After 15 seconds, the angle of elevation changes to 30°. If the aeroplane is flying at a height of 3000 m, find the speed of the aeroplane.
With reference to the given figure, a man stands on the ground at point A, which is on the same horizontal plane as B, the foot of the vertical pole BC. The height of the pole is 10 m. The man's eye is 2 m above the ground. He observes the angle of elevation of C, the top of the pole, as x°, where tan x° = `2/5`.
Calculate :
- the distance AB in metres;
- angle of elevation of the top of the pole when he is standing 15 metres from the pole. Give your answer to the nearest degree.
A man on the top of vertical observation tower observes a car moving at a 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 the car reach the observation tower? (Give your answer correct to nearest seconds).
The angles of elevation of the top of a tower from two points A and B at a distance of a and b respectively from the base and in the same straight line with it are complementary. Prove that the height of the tower is `sqrt(ab)`.
The angle of elevation of a cloud from a point 200 metres above a lake is 30° and the angle of depression of its reflection in the lake is 60°. Find the height of the cloud.
