SOLUTION: From a point A on level ground, the angle of elevation to the top of a tree is 38 degrees. From point B that is 46 feet farther from the tree, the angle of elevation is 22 degree

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Question 1198040: From a point A on level ground, the angle of elevation to the top of a tree is 38 degrees.
From point B that is 46 feet farther from the tree, the angle of elevation is 22 degrees. What is the height of the tree?
One of the possible answers are below:
A) 34.1 feet
B) 35.8 feet
C) 36.7 feet
D) 37.2 feet
E) 38.5 feet
Found 4 solutions by josgarithmetic, Edwin McCravy, math_tutor2020, MathTherapy:
Answer by josgarithmetic(39617) About Me  (Show Source):
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Draw the figure described. Let x be distance from point A to bottom of tree. Let y be how tall the tree.

system%28y%2F%28x%2B46%29=tan%2822%29%2Cy%2Fx=tan%2838%29%29

Make the substitution for x and solve the resulting equation in terms of y,... for the value.

Answer by Edwin McCravy(20054) About Me  (Show Source):
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A and B are the two points from which the given angles of elevation
are measured. G is the base of the tree, and T is the top of the tree.
x is the distance from A to the base of the tree G.

Using right triangle ATG,


TG%2FAG=opposite%2Fadjacent=y%2Fx=tan%2838%5Eo%29

Using right triangle BTG,


TG%2FBG=opposite%2Fadjacent=TG%2F%28BA%2BAG%29=y%2F%2846%2Bx%29=tan%2822%5Eo%29

So the system to solve is

system%28y%2F%28x%2B46%29=tan%2822%5Eo%29%2Cy%2Fx=tan%2838%5Eo%29%29

which the other tutor gave.

Solving is easier if you let tan%2838%5Eo%29=u and tan%2822%5Eo%29=v

system%28y%2F%28x%2B46%29=v%2Cy%2Fx=u%29

Cross multiply:

system%28y=v%28x%2B46%29%2Cy=ux%29

system%28y=vx%2B46v%2Cy=ux%29

Solve the 2nd equation for x

x=y%2Fu

Substitute that for x in the 1st equation:

y=v%28y%2Fu%29%2B46v

Multiply through by u

uy=vy%2B46uv

Get the y terms on the left:

uy-vy=46uv

Factor out y:

y%28u-v%29=46uv

y=%2846uv%29%2F%28u-v%29

Substituting tan(38o) for u and tan(22o) for v

y=matrix%281%2C2%2C38.48904651%2Cfeet%29

Edwin

Answer by math_tutor2020(3816) About Me  (Show Source):
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Here's a slight alternative approach to what the tutor Edwin McCravy has written.
I'll be using his diagram and the notation he set up of u = tan(38), v = tan(22)

The slight different approach is to let
BG = x
BA = 46
AG = x-46

tan(angle) = opposite/adjacent
tan(angle TBG) = TG/BG
tan(22) = y/x
y = x*tan(22)

tan(38) = y/(x-46)
tan(38) = x*tan(22)/(x-46) .... plug in y = x*tan(22)
u = x*v/(x-46) .... make replacements for u and v
u(x-46) = xv
ux - 46u = xv
ux-xv = 46u
x(u-v) = 46u
x = 46u/(u-v)
x = 46*tan(38)/(tan(38)-tan(22))
x = 95.263733005284, which is the approximate length of segment BG.

y = x*tan(22)
y = 95.263733005284*tan(22)
y = 38.489046505093
y = 38.5, which is the approximate length of segment TG.

Answer: E) 38.5 feet

---------------------------------------------------------------------------

Yet another approach

Refer to the diagram Edwin McCravy has drawn.

Angle TAG = 38 degrees.
angle TAB = 180-angle TAG = 180-38 = 142 degrees
This is angle A of triangle TAB.

Focus on triangle TAB
The interior angles T, A, B must add to 180 degrees.
T + A + B = 180
T + 142 + 22 = 180
T + 164 = 180
T = 180 - 164
T = 16

Use the law of sines to find side 'a' which is opposite angle A.

sin(A)/a = sin(T)/t
sin(142)/a = sin(16)/46
46*sin(142) = a*sin(16)
a = 46*sin(142)/sin(16)
a = 102.74524576336
This is the approximate length of segment TB.

Then focus on triangle TBG to say the following:
sin(angle) = opposite/hypotenuse
sin(angle TBG) = TG/TB
sin(22) = y/102.74524576336
y = 102.74524576336*sin(22)
y = 38.489046505093
y = 38.5


Answer: E) 38.5 feet

Answer by MathTherapy(10552) About Me  (Show Source):
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.
As ∡CAD = 38o, ∡CAB = 180 - 38 = 142o
In ΔABC, ∡BCA = 180 - (142 + 22), or 38 - 22 = 16o
Use Law of Sines to find AC, as follows:  
                                      AC * sin 16o = 46 * sin 22o ------ Cross-multiplying
                                               matrix%281%2C3%2C+AC%2C+%22=%22%2C+%2846+%2A+sin+%2822%5Eo%29%29%2Fsin+%2816%5Eo%29%29
                                               Continue solving for AC

                                 We then have: matrix%281%2C5%2C+sin+%28CAD%29%2C+%22=%22%2C+O%2FH%2C+%22=%22%2C+CD%2FAC%29
                                              matrix%281%2C3%2C+sin+%2838%5Eo%29%2C+%22=%22%2C+CD%2FAC%29
Since AC is already known (from above), you need to CONTINUE onward and solve for CD, the height of the tree. 

When done, you should get a height of approximately 38.49115544, which when rounded to 1 decimal place, is about 38.5'(CHOICE E.).