A ranger wants to determine the height of a tree across a river. Elevation from bank 50°. Walls back 37 feet. New angle of elevation is 26°.?
回答 (7)
2018-02-20 9:18 pm
Refer to the diagram below :
AB is the tree.
H is the first observing point, and K is the second observing point.
In ΔABH :
tan50° = h/d
d = h/tan50° …… [1]
In ΔABK :
tan26° = h/(d + 37)
d + 37 = h/tan26° …… [2]
[2] - [1] :
37 = (h/tan26°) - (h/tan50°)
h [(1/tan26°) - (1/tan50°)] = 37
h = 37 / [(1/tan26°) - (1/tan50°)]
h = 30.5
Height of the tree = 30.5 ft
AB is the tree.
H is the first observing point, and K is the second observing point.
In ΔABH :
tan50° = h/d
d = h/tan50° …… [1]
In ΔABK :
tan26° = h/(d + 37)
d + 37 = h/tan26° …… [2]
[2] - [1] :
37 = (h/tan26°) - (h/tan50°)
h [(1/tan26°) - (1/tan50°)] = 37
h = 37 / [(1/tan26°) - (1/tan50°)]
h = 30.5
Height of the tree = 30.5 ft
2018-02-20 6:30 pm
Always draw a diagram
Height of tree is x, original distance is d
tan 50 = x/ d .... or x = d tan 50
tan 26 = x/ d + 37 .... or x = (d +37) tan 26
eliminate x by making terms equal to each other
1.192d = 0.488d + 18.056
0.704d = 18.056
distance = 25.63
tan50 * 25.63 = x
1.192* 25.63 = 30.5
Ht of tree = 30.5 ft
You don't need to find the distance, but it is easier to see what is going on.
Height of tree is x, original distance is d
tan 50 = x/ d .... or x = d tan 50
tan 26 = x/ d + 37 .... or x = (d +37) tan 26
eliminate x by making terms equal to each other
1.192d = 0.488d + 18.056
0.704d = 18.056
distance = 25.63
tan50 * 25.63 = x
1.192* 25.63 = 30.5
Ht of tree = 30.5 ft
You don't need to find the distance, but it is easier to see what is going on.
2018-02-20 6:36 pm
h/d = tan50
dtan50/(d+37) = tan26
d = 25.633
h = 30.548 ft
dtan50/(d+37) = tan26
d = 25.633
h = 30.548 ft
2018-02-20 6:29 pm
Question seems to have something missing. How wide is the river?
If the tree is across the river, then this becomes a 2nd unknown ... well, maybe it will cancel out.
use tan (angle) = y/x ... w = width of river. ... y = height of tree (ignoring any height of ranger
Tan 50 = y/w ...... w tan 50 = y <<<< sub. below
Tan 26 = y/(w + 37) ... (w + 37) tan 26 = y
===========================================
w tan 50 = w tan 26 + 37 tan 26
w tan 50 - w tan 26 = 37 tan 26
w = (37 tan 26) / [tan 50 - tan 26] = 25.633
y = w tan 50 = 30.548 feet <<<< answer
If the tree is across the river, then this becomes a 2nd unknown ... well, maybe it will cancel out.
use tan (angle) = y/x ... w = width of river. ... y = height of tree (ignoring any height of ranger
Tan 50 = y/w ...... w tan 50 = y <<<< sub. below
Tan 26 = y/(w + 37) ... (w + 37) tan 26 = y
===========================================
w tan 50 = w tan 26 + 37 tan 26
w tan 50 - w tan 26 = 37 tan 26
w = (37 tan 26) / [tan 50 - tan 26] = 25.633
y = w tan 50 = 30.548 feet <<<< answer
2018-02-20 10:41 pm
tan(50)=h/d
tan(26)=h/(37+d)
tan(50)d=h
tan(26)(37+d)=h
tan(50)d=tan(26)(37+d)
tan(50)/tan(26)d=(37+d)
2.44d=37+d
1.44d=37
d=25.7 ft
tan(50)=h/d
tan(50)=h/25.7
h=30.6 ft (rounded)
tan(26)=h/(37+d)
tan(50)d=h
tan(26)(37+d)=h
tan(50)d=tan(26)(37+d)
tan(50)/tan(26)d=(37+d)
2.44d=37+d
1.44d=37
d=25.7 ft
tan(50)=h/d
tan(50)=h/25.7
h=30.6 ft (rounded)
2018-02-20 9:15 pm
tan(50) = h/d
h = d tan(50)
tan(26) = h/(d+37)
tan(26) = d tan(50) / (d+37)
cross multiply
(d+37) tan(26) = d tan(50)
d(tan(26)-tan(50)) = -37 tan(26)
d = -37 tan(26) /(tan(26)-tan(50))
d = 25.633
h = d tan(50)
h = (25.633)tan(50)
h = 30.548
https://gyazo.com/2ad655173d74ee8bb18c91bc75e16a3e
h = d tan(50)
tan(26) = h/(d+37)
tan(26) = d tan(50) / (d+37)
cross multiply
(d+37) tan(26) = d tan(50)
d(tan(26)-tan(50)) = -37 tan(26)
d = -37 tan(26) /(tan(26)-tan(50))
d = 25.633
h = d tan(50)
h = (25.633)tan(50)
h = 30.548
https://gyazo.com/2ad655173d74ee8bb18c91bc75e16a3e
2018-02-20 6:32 pm
as the distance to the tree from a fixed point cannot be measured, as the river width is unknown, a simple solution is not possible, otherwise by sighting the top of the tree, from a 45 degree angle at ground level, and measuring the distance to the base of the tree will give its height, as the tangent of 45 Degrees is 1. you do not say if one bank of the river is above the other.
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