Multiple choice

$n$ poles standing at equal distances on a straight road subtend the same angle $\alpha$ at a point $O$ on the road. If the height of the largest pole is $h$ and the distance of the foot of the smallest pole from $O$ is $a$, the distance between two consecutive poles is:

  1. ${\dfrac{h\sin\alpha-a\cos\alpha}{(n-1)\sin\alpha}}$
  2. $\displaystyle \dfrac{h\cos\alpha-a\cos\alpha}{(n-1)\cos\alpha}$
  3. $\displaystyle \dfrac{h\cos\alpha-a\sin\alpha}{(n-1)\sin\alpha}$
  4. $\displaystyle \dfrac{h\sin\alpha-a\cos\alpha}{(n-1)\cos\alpha}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the poles be at distances x1, x2, ..., xn from O. tan(alpha) = h/xn. Also, the poles are at equal distances d. xn = a + (n-1)d. The problem states they subtend the same angle alpha, which is only possible if they are at specific distances. The derivation leads to the formula in C.

AI explanation

Let the distance between consecutive poles be d, making the distance of the largest (nth) pole from O equal to a + (n-1)d. In the right triangle formed at O, the height is h and the base is this total distance, so tan alpha = h / (a + (n-1)d). Rearranging this equation gives a + (n-1)d = h cot alpha, which further simplifies to (n-1)d = h cot alpha - a. Since cot alpha = cos alpha / sin alpha, factoring out 1/sin alpha yields the distance between the poles as (h cos alpha - a sin alpha) / ((n-1) sin alpha).

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