Question 1109678: a pole tilts towards the sun in an angle of 10º from the vertical and cast 25ft shadow. The angle of elevation from the tip of the shadow to the top of the pole is 45º. how long is the pole?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! check out my diagram and you'll see what i'm doing.
the 10 degree angle is exaggerated so you can see clearly the relationship.
otherwise it would be too difficult to show the angles in the tight space that would have occurred.
your pole is 10 degrees off the vertical.
if you let CE be the vertical, then angle ECA is 10 degrees.
if you drop another perpendicular from A to D, and make EC the same size as AD, then AECD is a rectangle.
being a rectangle, side AD is parallel and congruent to EC and side AE is parallel and cngruent to DC.
AC, which is the pole, is a diagonal of this rectangle.
since alternate interior angles of parallel lines are congruent, then angle ECA is congruent to angle CAD.
triangle ACD is a right triangle.
triangle ABD is also a right triangle.
since AC is the hypotenuse of triangle ACD and AB is the hypotenuse of triangle ABD, and AD is the vertical leg of both these triangles, we can use this fact to find the length of AC.
first we want to find the length of AD.
angle B is 45 degrees.
sine (B) is equal to opposite / hypotenuse which is equal to AD / AB.
since B is 45 degrees and AB is 25, this equation becomes:
sine (45) = AD / 25
solve for AD to get AD = 25 * sine (45) = 17.67766953.
now we go to triangle ACD.
that is also a right triangle.
sine(ACD) = opposite / hypotenuse = AD / AC.
since angle ACD is 80 degrees and AD is equal to 17.67766953, the equation becomes:
sine (80) = 17.67766953 / AC.
solve for AC to get AC= 17.67766953 / sine (80) which is equal to 17.95037608.
that's the length of the pole.
we could also have used the law of sines to find the length of AC.
we know that a / sine (A) = b / sine (B) = c / sine (C).
we know that angle B in triangle ABC is equal to 45 degrees.
since angle ACD is 10 degrees off the vertical, then we know that angle ACB is equal to 90 + 10 = 100 degrees.
we know that angle ACB can be called angle C if we are just restricting our view to triangle ABC.
we know that angle A in triangle ABC is equal to 180 - 100 - 45 which makes angle A equal to 35 degrees.
in triangle ABC we have:
angle A = 35 degrees.
angle B = 45 degrees.
angle C= 100 degrees.
we label side AB = c because it's opposite angle C.
we label side AC = b because it's opposite angle B.
we label side CB = a because it's opposite angle A.
by the law of sinces:
a / sine (A) = b / sine (B) = c / sine (C)
we know that c is equal to 25, therefore we can use c / sine (C) to find the length of b.
the part of the law of sines formula we use is:
b / sine (B) = c / sine (C)
we know that angle B = 45 degrees and we know that angle C = 100 degrees and we know that c = 25, so the formula becomes:
b / sine (45) = 25 / sine (100)
solve for b to get b = 25 * sine (45) / sine (100)
this makes b = 17.95037608.
that's what we calculated before, so we got the same answer two ways.
the first was by construction of right triangles.
the second was by use of the law of sines.
looks like 17.95037608 feet is the length of your pole.
here's my diagram.
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