SOLUTION: From a point, a point on the tower 20 m above the ground was observed .By doubling the angle of elevation another point of the tower which was 25 m vertically above the first was o

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Question 1068467: From a point, a point on the tower 20 m above the ground was observed .By doubling the angle of elevation another point of the tower which was 25 m vertically above the first was observed.Find the distance between the foot of the tower and point of observation
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let X be the distance between the observer and the tower,
tan%28theta%29=20%2FX
X=20%2Ftan%28theta%29
.
.
tan%282%2Atheta%29=45%2FX
X=45%2Ftan%282%2Atheta%29
.
.
20tan%28theta%29=45%2Ftan%282%2Atheta%29
20tan%282%2Atheta%29=45tan%28theta%29
Using an identity,
20%28%282tan%28theta%29%29%2F%281-tan%5E2%28theta%29%29%29=45tan%28theta%29
Use a substitution for better readability,
%2840u%29%2F%281-u%5E2%29=45u
40u=45u-45u%5E3
45u%5E3-5u=0
5u%289u%5E2-1%29=0
Only the second solution is helpful,
9u%5E2-1=0
9u%5E2=1
u%5E2=1%2F9
u=1%2F3
tan%28theta%29=1%2F3
So then,
X=20%2Ftan%28theta%29
X=20%2F%281%2F3%29
X=60m