Question 1203114: Santa has just left presents at Ms. Cane’s house and is heading for Mr.John’s place. From his sleigh Santa can see Ms. Cane’s chimney at an angle of depression of 5.3º and Mr. John at 10.2º. If Ms. Cane and Mr. John lives 25 kilometers apart, calculate Santa’s altitude.
Found 3 solutions by josgarithmetic, math_tutor2020, ikleyn:
Answer by josgarithmetic(39618)   (Show Source):
You can put this solution on YOUR website! We might assume that we could ignore how high is the chimneys.
Then draw the diagram described.
Two right triangles with a common corner point; left triangle with leg x, leg y, angle at the shared point 5.3 deg. right side triangle with leg 25-x, leg y, angle at the common point 10.2 degree.
The question would be asking for y.
Solve for y.
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The algebra or arithmetic steps give  ;
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Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
Diagram
Points- A = the top of Ms. Cane's chimney
- B = the top of Mr. John's chimney
- C = Santa's location
- D and E = needed to form right triangles ACD and BCE respectively.
- F = point directly under point C, and the same horizontal level as A and B
Angles- angle ACD = 5.3 degrees (one of the angles of depression)
- angle BCE = 10.2 degrees (the other angle of depression)
- angle CAF = 5.3 degrees
- angle CBF = 10.2 degrees
Segments- AF = x
- FB = 25-x
- AB = 25 kilometers
- CF = h = unknown height or altitude
The goal is to calculate the value of h.
An angle of depression is where you start looking at the horizon, then you move your viewpoint downward that number of degrees until reaching the target.
This explains how angles ACD and BCE are set up.
Angle CAF is congruent to angle ACD because of the alternate interior angle theorem. AB is parallel to DE since both are horizontal.
Also, angle CBF = angle BCE for similar reasoning.
Focus on right triangle ACF
tan(angle) = opposite/adjacent
tan(angle CAF) = CF/AF
tan(5.3) = h/x
h = x*tan(5.3)
We'll use this in a substitution step later on.
Now focus on right triangle BCF
tan(angle) = opposite/adjacent
tan(angle CBF) = CF/FB
tan(10.2) = h/(25-x)
tan(10.2) = x*tan(5.3)/(25-x) ...... Substitution: replace h with x*tan(5.3)
tan(10.2)*(25-x) = x*tan(5.3)
25*tan(10.2)-x*tan(10.2) = x*tan(5.3)
25*tan(10.2) = x*tan(5.3)+x*tan(10.2)
25*tan(10.2) = x*( tan(5.3)+tan(10.2) )
x = 25*tan(10.2)/( tan(5.3)+tan(10.2) )
x = 16.4953525945108
This value is approximate.
Make sure your calculator is in degree mode.
That x value leads to
h = x*tan(5.3)
h = 16.4953525945108*tan(5.3)
h = 1.5302275941042
This value is approximate.
Santa's altitude or height is approximately 1.5302 km
Round this value however your teacher instructs.
Extra info
1.5302 km = 5020.34121 feet (approximate)
1.5302 km = 0.9508222 miles (approximate)
Answer by ikleyn(52788)  (Show Source):
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