SOLUTION: A flagpole is perpendicular to the horizontal but is on a slope that rises 8.6° from the horizontal. The pole casts a 70-foot shadow down the slope and angle of elevation of the s

Algebra ->  Triangles -> SOLUTION: A flagpole is perpendicular to the horizontal but is on a slope that rises 8.6° from the horizontal. The pole casts a 70-foot shadow down the slope and angle of elevation of the s      Log On


   

Question 1199090: A flagpole is perpendicular to the horizontal but is on a slope that rises 8.6° from the horizontal. The pole casts a 70-foot shadow down the slope and angle of elevation of the sun measured from the slope is 24.3°. How tall is the pole? Round your answer to the nearest 0.1 foot.
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

"A flagpole is perpendicular to the horizontal" is another way of saying "the flagpole is vertical".

One way to draw out the diagram:


Points:
A = base of the hill
B = tip of the shadow
C = base of the flagpole
D = top of the flagpole
E = point used to help label the 8.6 degree angle of elevation

Given info:
angle EAB = 8.6 degrees
angle DBC = 24.3 degrees
segment BC = 70 ft = shadow length
The flagpole itself is in red (segment CD).

Next we draw a horizontal line through point C.
Mark points F and G on the new horizontal line and line AC respectively.

Because AE and CF are horizontal parallel lines, we know that
angle EAB = angle FCG = 8.6 degrees
and this is because of the corresponding angles theorem.

Mark point H somewhere to the left of C on the line CF.
I'll place H as the intersection of CF and BD.
It doesn't matter where H is as long as it's to the left of C.

The introduction of point H is to help label the angle HCB which is congruent to angle FCG
angle HCB = angle FCG = 8.6 degrees
since these are vertical angles.

This is the updated diagram with those items added

Then,
angle BCD = (angle HCB)+(angle HCD)
angle BCD = (8.6)+(90)
angle BCD = 98.6 degrees
This is angle C of triangle BCD.

Keeping our focus on triangle BCD only, we can say:
B+C+D = 180
24.3+98.6+D = 180
122.9+D = 180
D = 180-122.9
D = 57.1
This is angle BDC

We'll still keep our focus on triangle BCD only.
Apply the law of sines to find side 'b', aka segment CD.

b/sin(B) = d/sin(D)
b/sin(24.3) = 70/sin(57.1)
b = sin(24.3)*70/sin(57.1)
b = 34.308389211743
b = 34.3
Therefore, segment CD is roughly 34.3 feet long.

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Answer: The flagpole is approximately 34.3 feet tall.

This is roughly comparable to about halfway between a 3 and 4 story building, assuming each floor is 10 ft high.