Try the tangent!
Geometry
Level
3
A pedestal (height a) sustents a column (height b, b>a). How far from the monument should an observer be so he can see the pedestal and the column under equal angles?
After you find the answer, evaluate it for a=4 and b=5.
The answer is 12.
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Calculating tangents from the angles:
I) tan(theta) = a/x
II) tan(2theta) = (a+b)/x
2tan(theta)/(1-tan²(theta) = (a+b)/x
(2a/x)/(1 - a²/x²) = (a+b)/x
(2a/x)/[(x²-a²)/x²] = (a+b)/x
2ax²/x[x²-a²] = (a+b)/x
2ax² = (a+b)(x²-a²)
2ax² = (a+b)x² - (a+b)a²
(a+b)x² - 2ax² = (a+b)a²
(b-a)x² = (a+b)a²
x² = a²(a+b)/(b-a)
x = a[(b+a)/(b-a)]^(1/2).
Then, for a=4 and b=5, x = 49^(1/2) = 4*3 = 12.