A flagstaff stands in the centre of a rectangular
field whose diagonal is 1200 m, and subtends angles
15° and 45° at the mid points of the sides of the field. The
height of the flagstaff is

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$300 \sqrt{2 - \sqrt{3}} m$

400 m

200 m

$300 \sqrt{2 + \sqrt{3}} m$

answer is C.

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Detailed Solution

Let $O P$ be the flagstaff of height $h$ standing
at the centre $O$ of the rectangular field $A B C D$ subtending
angles 15° and 45° at $E$ and $F$ the mid points of the sides
$A D$ and $D C$ of the field.(Fig. 27.9)

then $O E = h c o t 15 °$ .

$= h (2 + \sqrt{3})$

and $O F = h \cdot \cot 45^{\circ} = h$

$\begin{array}{l} \Rightarrow E F = h \sqrt{1 + (2 + \sqrt{3})^{2}} \\ = 2 h \sqrt{2 + \sqrt{3}} \\ \Rightarrow 1200 = A C = 2 E F = 4 h \sqrt{2 + \sqrt{3}} \\ \Rightarrow h = \frac{300}{\sqrt{2 + \sqrt{3}}} = 300 \sqrt{2 - \sqrt{3}} . \end{array}$