SOLUTION: Because of prevailing winds, a tree grew so that it was leaning 4 degrees from vertical. At a point 35 meters from the tree the angle of elevation to the top of the tree is 23 degr

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Question 1029851: Because of prevailing winds, a tree grew so that it was leaning 4 degrees from vertical. At a point 35 meters from the tree the angle of elevation to the top of the tree is 23 degrees. Find the height of the tree.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the diagram i worked with is shown here.
look below the diagram for the discussion that follows.
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the right triangle formed is ACD.
the triangle formed that goes to the height of the tree is ABD.
drop a perpendicular from B to E on the line AD.
you have two right triangles formed.
they are ABE and DBE.
angle BAE is 23 degrees.
angle ABE is 67 degrees.
angle AEB is 90 degrees.
angle BDC is 4 degrees because this is the angle from the vertical that the tree is slanted at.
since angle CDE is a right angle, then angle BDE is 86 degrees.
since triangle DBE is a right triangle, than angle DBE is 4 degrees.
line BE is the altitude of both triangles ABE and DBE.
the line AD is broken up into two parts.
the first part is line AE whose length is designated as x.
the second part is line ED whose length is designated as 35-x.
AE and ED together form the line AD which is given as 35 meters long.
the length of the line BE is designated as h.

from this we derive the following:

tan(23) = h/x

tan(86) = h/(35-x)

solve for h in both these formulas and you get:

h = x * tan(23)
h = (35-x) * tan(86)

this gets you x * tan(23) = (35-x) * tan(86)
simplify this equation to get x * tan(23) = 35 * tan(86) - x * tan(86)
add x * tan(86) to both sides of this equation to get x * tan(23) + x * tan(86) = 35 * tan(86).
factor out the x to get x * (tan(23) + tan(86)) = 35 * tan(86).
divide both sides of this equation by (tan(23) + tan(86)) to get:
x = (35 * tan(86) / (tan(23) + tan(86)).
this results in x = 33.99107122.
this makes 35 - x = 1.008928777.

h is calculated to be 33.99107122 * tan(23) and h is also calculated to be 1.008928777 * tan(86).

both formulas tell you that h is equal to 14.42835371 meters.

that's the height of the tree from the ground which is the length of line BE in the diagram.

i believe the actual height of the tree would be the length of the line BD.

that measures the length from the base of the tree to the top of the tree.

that can be calculated based on the following formulas.

sin(86) = BE / BD.
solve for BD to get BD = BE / sin(86).

cos(86) = ED / BD.
solve for BD to get BD = ED / cos(86).

BE^2 + ED^2 = BD^2 which makes BD = sqrt(BE^2 + ED^2).

all of these point to the same length of the line BD which is equal to 14.46358628 meters.

the vertical height of the tree is equal to 14.42835371 meters which is the length of the line BE.

what i think is the actual height of the tree is equal to 14.46358628 meters which is the length of the line BD.

that's what i get.