Multiple choice

$AB$ is a straight road leading to $C$, the foot of a tower . $A$ is at a distance $200$ meters from $C$ and $B$ at $75$ meters from $C$. If the angle of elevation of the tower at B be double the angle of elevation at A, then the height of the tower is

  1. $125m$
  2. $120m$
  3. $115m$
  4. $100m$
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D Correct answer
Explanation

Let h be the height of the tower. tan(theta) = h/200 and tan(2*theta) = h/75. Using the identity tan(2*theta) = 2*tan(theta) / (1 - tan^2(theta)), we get h/75 = 2(h/200) / (1 - (h/200)^2). Solving for h gives h^2 = 10000, so h = 100.

AI explanation

Using the right triangle formulas, if the angle of elevation at A is theta, then tan theta equals h over 200. Because the angle of elevation at B is double that at A, tan 2theta equals h over 75. Using the double angle identity, tan 2theta equals 2 tan theta divided by (1 minus tan squared theta), we substitute the first equation into the second to get h over 75 equals 2 multiplied by (h over 200) divided by (1 minus (h over 200) squared). Solving this equation for h gives h over 75 equals h over 100 divided by (1 minus h squared over 40000), which simplifies to 40000 minus h squared equals 30000, resulting in a height of 100 m.

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