Multiple choice

A balloon of radius r makes an angle at the eye of an observer and the angle of elevation of its centre is $\beta$. The height of its centre from the ground level is given by

  1. $r cos \dfrac {\beta}{2}sec\alpha$
  2. $r cos \beta sec\dfrac {\alpha}{2}$
  3. $r sin \dfrac {\alpha}{2}cosec\beta$
  4. $r sin\beta cosec\dfrac {\alpha}{2}$
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D Correct answer
Explanation

Using trigonometry in the right triangle formed by the observer, the center of the balloon, and the ground, the height h = d * sin(beta). The angle alpha subtended at the eye relates to the radius r as sin(alpha/2) = r / d. Thus d = r / sin(alpha/2). Substituting d gives h = r * sin(beta) * cosec(alpha/2).

AI explanation

The balloon of radius r subtends an angle alpha at the observer's eye, so the line of sight to the center and the tangent to the balloon form a right triangle where sin(alpha / 2) = r / d, giving the distance to the center d = r * cosec(alpha / 2). The height of the center is found using the angle of elevation beta, so h = d * sin(beta). Substituting d into the height equation gives h = r * sin(beta) * cosec(alpha / 2). The result is r * sin(beta) * cosec(alpha / 2).

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