SOLUTION: A pole casts a shadow of 25ft at one time and a shadow of 10 ft at a later time when the angle of elevation is twice as large. Find the height of the pole, to the nearest foot.

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Question 415189: A pole casts a shadow of 25ft at one time and a shadow of 10 ft at a later time when the angle of elevation is twice as large. Find the height of the pole, to the nearest foot.
I have tried using the double angle identity of tan and am stuck at 50h/625-h squared = h/10
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
here's how i solved it.

x = angle of elevation.
y = height of pole.

height of pole is equal to y

tan(x) = y/25

tan(2x) = y/10

the double angle tan formula is:

tan(2x) = 2tan(x) / (1 − tan^2(x))

we know that tan(2x) = y/10
we know that tan(x) = y/25

substituting in tan(2x) = 2tan(x) / (1 − tan^2(x)), we get:

y/10 = (2y/25) / (1 - (y/25)^2)

simplify to get:

y/10 = (2y/25) / (1 - (y^2/625))

multiply both sides of the equation by (1 - (y^2/625) and divide both sides of the equation by (y/10) to get:

1 - (y^2/625) = (2y/25) / (y/10)

this is equivalent to:

1 - (y^2/625) = (2y/25) * (10/y)

simplify to get:

1 - (y^2/625) = 20/25

add (y^2/625) to both sides of the equation and subtract 20/25 from both sides of the equation to get:

1 - (20/25) = y^2/625

simplify to get:

5/25 = y^2/625

multiply both sides of the equation by 625 to get:

(625*5)/25 = y^2

simplify to get:

125 = y^2

take square root of both sides of the equation to get:

y = +/- 11.18033989

you have tan(x) = 11.18033989/25 = .447213596

arctan (.447213596) = 24.09484255 degrees = x.

you also have tan(2x) = 11.18033989/10 = 1.118033989

arctan(1.118033989) = 48.1896851 degrees = 2x

the results check out ok.

y = 11.18033989 feet which is the height of the pole.

formulas we started with are:

y = 25 * tan(x) and y = 10 * tan(2x)

our angles became:

x = 24.09484255 degrees
2x = 48.1896851 degrees

tan(x) = .447213596
tan(2x) = 1.118033989

y = 25*tan(x) = 25 * .447213596 = 11.18033989
y = 10*tan(2x) = 10 * 1.118033989 = 11.18033989

height of the pole is 11.18033989 feet.