Multiple choice

Let $10$ vertical poles standing at equal distances on a straight line, subtend the same angle of elevation $\alpha$ at a point $O$ on this line and all the poles are on the same side of $O$. If the height of the longest pole is $h$ and the distance of the foot of the smallest pole from $O$ is $a$; then the distance between two consecutive poles, is

  1. $\displaystyle \frac{h \sin \alpha + a \cos \alpha}{9 \cos \alpha}$
  2. $\displaystyle \frac{h \cos \alpha - a \sin \alpha}{9 \sin \alpha}$
  3. $\displaystyle \frac{h \sin \alpha + a \cos \alpha}{9 \sin \alpha}$
  4. $\displaystyle \frac{h \cos \alpha - a \sin \alpha}{9 \cos \alpha}$
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B Correct answer
Explanation

Let the poles be at distances x1, x2, ..., x10 from O. tan(alpha) = h_i / x_i. x_i = h_i / tan(alpha). The distance between consecutive poles is (h_{i+1} - h_i) / tan(alpha). Using the geometry of the setup, the formula simplifies to the given expression.

AI explanation

Let the distance between two consecutive poles be d, making the distance of the foot of the longest pole from O equal to a + 9d. Since the longest pole has a height of h and subtends an angle of alpha, we have tan(alpha) = h / (a + 9d). Rearranging this equation to solve for d gives a + 9d = h / tan(alpha). Substituting tan(alpha) with sin(alpha)/cos(alpha) and isolating 9d results in 9d = (h cos(alpha) - a sin(alpha)) / sin(alpha). Dividing by 9 gives the distance between two consecutive poles as (h cos(alpha) - a sin(alpha)) / (9 sin(alpha)).

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