Question

the sun is at an angle of elevation of 75 degrees in the sky at this time a building's shadow is only 15m long how tall is the building

Answer 1

$Let \: height \:of \:the \: building (AB) = h \:m$

$Length \:of \: the \:shadow (CB) = 15 \:m$

$In \: \triangle ABC , we \:have$

$tan 75 \degree = \frac{AB}{CB}$

$\implies tan( 45 + 30 ) = \frac{h}{15}$

/*

$tan 75 = tan(45+30) \\= \frac{tan45 + tan 30 }{1 - tan 45 tan 30 }$

$= \frac{1 + \frac{1}{\sqrt{3}}}{1- \frac{1}{\sqrt{3}}}\\= \frac{\sqrt{3}+1}{\sqrt{3}-1}$

$= \frac{(\sqrt{3}+1)(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)}\\= \frac{ 3+1+2\sqrt{3}}{3-1} \\= \frac{2(2+\sqrt{3}}{2}\\= 2+\sqrt{3} \\= 2 + 1.732 = 3.732 \: --(1)$

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$\implies 3.732 = \frac{h}{15}$

$\implies 15 \times 3.732 = h$

$\implies h = 55.98 \: m$

Therefore.,

$\red {Height \:of \:the \: building}\green {= 55.98 \:m }$

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