An aeroplane 4500 m high passes vertically above another plane at an instant when the angles of elevation of the two planes from a point P on the ground are 60° and 30°. Find the vertical distance between the planes.
Answers
Given :
An aeroplane 4500 m high passes vertically above another plane at an instant when the angles of elevation of the two planes from a point P on the ground are 60° and 30°.
To Find: The vertical distance between the planes.
Solution: Let A and B the two planes
Height of plane A = 4500 m
P is a point on the ground such that the two planes make an angle of elevation as 60° and 30°.
AC = 4500 m
AB = h
BC = 4500 - h
PC = x
Now in right ∆ACP,
tan θ = AC/PC
tan 60° = 4500/x
√3 = 4500/x
x = 4500/√3 ……..(1)
Similarly, in right ∆BCP,
tan θ = BC/PC
tan 30° = (4500 - h) /x
1/√3 = 4500 - h /x
x = √3(4500 - h ) ……..(2)
From eq 1 and 2 , we get
4500/√3 = √3(4500 - h)
4500 = √3(4500 - h) × √3
4500 = 3(4500 - h)
4500 - h = 4500/3
4500 - h = 1500
h = 4500 - 1500
h = 3000 m
Hence, the vertical distance between the planes is 3000 m .
Learn more:
From a point P on the ground the angle of elevation of a 10 m tall building is 30°. A flag is hoisted at the top of the building and the angle of elevation of the top of the flag-staff from P is 45°. Find the length of the flag-staff and the distance of the building from the point P. (Take 3=1.732).
https://brainly.in/question/11911810
A parachutist is descending vertically and makes angles of elevation of 45° and 60° at two observing points 100 m apart from each other on the left side of himself. Find the maximum height from which he falls and the distance of the point where he falls on the ground form the just observation point.
https://brainly.in/question/11872435
ɢɪᴠᴇɴ :
ᴀɴ ᴀᴇʀᴏᴘʟᴀɴᴇ 4500 ᴍ ʜɪɢʜ ᴘᴀssᴇs ᴠᴇʀᴛɪᴄᴀʟʟʏ ᴀʙᴏᴠᴇ ᴀɴᴏᴛʜᴇʀ ᴘʟᴀɴᴇ ᴀᴛ ᴀɴ ɪɴsᴛᴀɴᴛ ᴡʜᴇɴ ᴛʜᴇ ᴀɴɢʟᴇs ᴏғ ᴇʟᴇᴠᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ᴛᴡᴏ ᴘʟᴀɴᴇs ғʀᴏᴍ ᴀ ᴘᴏɪɴᴛ ᴘ ᴏɴ ᴛʜᴇ ɢʀᴏᴜɴᴅ ᴀʀᴇ 60° ᴀɴᴅ 30°.
ᴛᴏ ғɪɴᴅ: ᴛʜᴇ ᴠᴇʀᴛɪᴄᴀʟ ᴅɪsᴛᴀɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ᴛʜᴇ ᴘʟᴀɴᴇs.
sᴏʟᴜᴛɪᴏɴ: ʟᴇᴛ ᴀ ᴀɴᴅ ʙ ᴛʜᴇ ᴛᴡᴏ ᴘʟᴀɴᴇs
ʜᴇɪɢʜᴛ ᴏғ ᴘʟᴀɴᴇ ᴀ = 4500 ᴍ
ᴘ ɪs ᴀ ᴘᴏɪɴᴛ ᴏɴ ᴛʜᴇ ɢʀᴏᴜɴᴅ sᴜᴄʜ ᴛʜᴀᴛ ᴛʜᴇ ᴛᴡᴏ ᴘʟᴀɴᴇs ᴍᴀᴋᴇ ᴀɴ ᴀɴɢʟᴇ ᴏғ ᴇʟᴇᴠᴀᴛɪᴏɴ ᴀs 60° ᴀɴᴅ 30°.
ᴀᴄ = 4500 ᴍ
ᴀʙ = ʜ
ʙᴄ = 4500 - ʜ
ᴘᴄ = x
ɴᴏᴡ ɪɴ ʀɪɢʜᴛ ∆ᴀᴄᴘ,
ᴛᴀɴ θ = ᴀᴄ/ᴘᴄ
ᴛᴀɴ 60° = 4500/x
√3 = 4500/x
x = 4500/√3 ……..(1)
sɪᴍɪʟᴀʀʟʏ, ɪɴ ʀɪɢʜᴛ ∆ʙᴄᴘ,
ᴛᴀɴ θ = ʙᴄ/ᴘᴄ
ᴛᴀɴ 30° = (4500 - ʜ) /x
1/√3 = 4500 - ʜ /x
x = √3(4500 - ʜ ) ……..(2)
ғʀᴏᴍ ᴇq 1 ᴀɴᴅ 2 , ᴡᴇ ɢᴇᴛ
4500/√3 = √3 (4500 - ʜ)
4500 = √3 (4500 - ʜ) × √3
4500 = 3(4500 - ʜ)
4500 - ʜ = 4500/3
4500 - ʜ = 1500
ʜ = 4500 - 1500
ʜ = 3000 ᴍ
ʜᴇɴᴄᴇ, ᴛʜᴇ ᴠᴇʀᴛɪᴄᴀʟ ᴅɪsᴛᴀɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ᴛʜᴇ ᴘʟᴀɴᴇs ɪs 3000 ᴍ .
