Multiple choice

The angle of elevation of the top of tower as observed from a pint on the horizontal ground is 'x' if we move a distance 'd' towards the foot of the tower the angle of elevation increases to 'y' then the height of the tower is

  1. $\displaystyle \frac{d\tan x\tan y}{\tan y-\tan x}$
  2. $\displaystyle d(\tan y+\tan x)$
  3. $\displaystyle d(\tan y-\tan x)$
  4. $\displaystyle \frac{d\tan x\tan y}{\tan y+\tan x}$
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A Correct answer
Explanation

Let h be height, y be the angle at distance x from tower, x be angle at distance x+d. tan(y) = h/x, so x = h/tan(y). tan(x) = h/(x+d). Substituting x: tan(x) = h / (h/tan(y) + d) = h*tan(y) / (h + d*tan(y)). Rearranging for h: h*tan(x) + d*tan(x)*tan(y) = h*tan(y). h(tan(y) - tan(x)) = d*tan(x)*tan(y). h = d*tan(x)*tan(y) / (tan(y) - tan(x)).

AI explanation

Let the height of the tower be h, the initial distance be d1, and the closer distance be d1 minus d. Using the initial angle x, tan(x) = h divided by d1, so d1 equals h divided by tan(x). At the closer point, tan(y) = h divided by (d1 minus d). Substituting d1 gives tan(y) = h divided by ((h divided by tan(x)) minus d). Solving this equation for h results in h = (d times tan(x) times tan(y)) divided by (tan(y) minus tan(x)).

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