Multiple choice

A flag staff of height $'h'$ stands in the centre of a rectangular field and subtends angle of $15^{o}$ and $45^{o}$ at the mid point of its sides. Find the distance between them

  1. $h\sqrt{2+\sqrt{3}}$
  2. $2h\sqrt{2+\sqrt{3}}$
  3. $4h\sqrt{2+\sqrt{3}}$
  4. $3h\sqrt{2+\sqrt{3}}$
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B Correct answer
Explanation

Let the distance from the center to the midpoints of the sides be x and y. Then h/x = tan(15) and h/y = tan(45). The distance between the midpoints is sqrt(x^2 + y^2). Using trigonometric identities for tan(15) = 2 - sqrt(3), the calculation leads to the result.

AI explanation

Using basic trigonometric ratios, the distances from the center of the field to the midpoints of the sides are d1 = h cot 15 and d2 = h cot 45, which simplifies to h(2 + sqrt(3)) and h respectively. These distances meet at the center at a right angle, so by the Pythagorean theorem, the distance between the midpoints is sqrt((h(2 + sqrt(3)))^2 + h^2). Factoring out h gives h sqrt(4 + 4 sqrt(3) + 3 + 1), which simplifies to h sqrt(8 + 4 sqrt(3)) or 2h sqrt(2 + sqrt(3)). The distance is 2h sqrt(2 + sqrt(3)).

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