Multiple choice

If the angles of depression and elevation of the top of a tower of height $h$ from the top and bottom of a second tower are $x$ and $y$ respectively then the height of the second tower is

  1. $\displaystyle h\left ( \cot y+\cot x \right )$
  2. $\displaystyle h\left ( \tan x+\tan y \right )$
  3. $\displaystyle h\left ( 1+\tan x\cot y \right )$
  4. $\displaystyle h\left ( \tan y\cot x +1\right )$
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C Correct answer
Explanation

Let the second tower have height H. The distance between towers is d. From the top of the second tower, tan(x) = h/d, so d = h/tan(x). From the bottom, tan(y) = H/d, so H = d * tan(y) = (h/tan(x)) * tan(y) = h * tan(y) * cot(x). Total height = h + H = h(1 + tan(y) * cot(x)).

AI explanation

Let the height of the second tower be H, and the horizontal distance between the towers be d. From the bottom of the second tower, the angle of elevation to the first tower is y, so tan(y) equals h divided by d, meaning d equals h divided by tan(y), or h times cot(y). From the top of the second tower, the depression angle x means the vertical difference between the towers is d times tan(x). Substituting d into this expression gives a vertical difference of h times tan(x) times cot(y). Adding this difference to the height of the second tower gives H = h + h(tan(x)cot(y)), which simplifies to h(1 + tan(x)cot(y)).

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