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)   (Show Source): You can put this solution on YOUR website!
Draw the figure described. Let x be distance from point A to bottom of tree. Let y be how tall the tree.



Make the substitution for x and solve the resulting equation in terms of y,... for the value.
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!


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,




Using right triangle BTG,




So the system to solve is



which the other tutor gave.

Solving is easier if you let  and 



Cross multiply:





Solve the 2nd equation for x



Substitute that for x in the 1st equation:



Multiply through by u



Get the y terms on the left:



Factor out y:





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



Edwin
Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

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)   (Show Source): You can put this solution on YOUR website!

.
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
                                               
                                               Continue solving for AC

                                 We then have: 
                                              
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.).
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