Multiple choice

The angle of elevation of an object from a point $P$ on the level ground is $\alpha$. Moving $d$ meters on the ground towards the object, the angle of elevation is found to be $\beta$, then the height (in meters) of the object is

  1. $ d\tan\alpha$
  2. $ d\cot\beta$
  3. $\dfrac d{\cot\alpha+\cot\beta}$
  4. $\dfrac d{\cot\alpha-\cot\beta}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let height be h. tan(alpha) = h/x and tan(beta) = h/(x-d). x = h/tan(alpha) = h*cot(alpha). x-d = h*cot(beta). d = x - (x-d) = h*cot(alpha) - h*cot(beta). h = d / (cot(alpha) - cot(beta)).

AI explanation

Let the total distance from the second observation point to the base of the object be x, making the height h = x * tan(beta) and also h = (x + d) * tan(alpha). Equating the two expressions for x gives h / tan(beta) = h / tan(alpha) + d. Rewriting the tangents as cotangents yields h * cot(beta) - h * cot(alpha) = d. Factoring out h gives h times the quantity (cot(beta) - cot(alpha)) equals d, so solving for h gives d divided by (cot(beta) - cot(alpha)). To match the given form, multiply the numerator and denominator by negative one to get d / (cot(alpha) - cot(beta)). The result is d / (cot(alpha) - cot(beta)).

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