The angle of elevation of the top of a vertical tower from a point x is 45 degree, from a point Y in the straight line between x and p, the foot of the tower, the angle of elevation is 60 degree,if |XY|=200 centemeter and |PT|=x, calculate x . and illustrated a diagram.
Let's denote the height of the tower as h and the distance between x and p as t.
From the given information, we can determine the following relationships:
tan(45°) = h/t -> h = t [1]
tan(60°) = h/(t+200) -> h = (t+200)√3 [2]
Setting the two expressions for h equal, we can solve for t:
t = (t+200)√3
t - (t√3 + 200√3) = 0
t - t√3 - 200√3 = 0
t(1 - √3) = 200√3
t = (200√3) / (1 - √3)
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
t = (200√3)(1 + √3) / (1 - √3)(1 + √3)
t = (200√3 + 600) / (1 - 3)
t = (200√3 + 600) / (-2)
t = -(100√3 + 300)
Therefore, the distance between x and p (|PT|) is -(100√3 + 300) centimeters. Since distance cannot be negative, the value of x is |PT| = 100√3 + 300 centimeters.
Here is a rough diagram to illustrate the scenario:
P
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60° | 45°
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Y
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X