Multiple choice

A man observes a tower AB of height h from a point P on the ground. He moves a distanced towards the foot of the tower and finds that the angle of elevation is doubled. He further moves a distance $3d/4$ in the same direction and finds that the angle of elevation is three times that at P then $36h^2 =35d^2$.

  1. True

  2. False
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A Correct answer
Explanation

Using the tangent function for angles at P, P', and P'', and the given distances, one can derive the relationship between h and d. The geometric constraints lead to the equation 35d^2 = 36h^2, confirming the statement is true.

AI explanation

Let the height be h and the initial distance to the tower be x, giving tan of the first angle equal to h divided by x. After moving distance d, the angle doubles, so tan of twice the angle equals h divided by the quantity x minus d, which yields the equation h equals x times d divided by x plus d. After moving another 3d divided by 4, the angle triples, meaning tan of three times the first angle equals h divided by the quantity x minus 7d divided by 4. By substituting h and applying the triple angle formula, the algebra simplifies to the relation 36 h squared equals 35 d squared, making the statement true.

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