Question

From the top of a house 32m high,the angle of elevation of the top of a tower is 45° and to the foot of a tower is 30°.Find the height of the tower.​

Answer 1

$Height \:of \:the \: House (AB) = 32 \:m \\\Let \:Height \:of \:the \:Tower (CE) = H\:m$

$Distance \: between \: foot \:of \: the \:house \\to \: foot \:of \:the \:Tower (BC) = x \:m$

$Let \: DE = y \:m$

$i) In \: \triangle ADE , \\tan \angle {DAE} = \frac{DE}{AD}$

$\implies tan 45\degree = \frac{y}{x}$

$\implies 1 = \frac{y}{x}$

$\implies x = y \: ---(1)$

$ii) In \: \triangle ADC , \\tan \angle {DAC} = \frac{DC}{AD}$

$\implies tan 30\degree = \frac{32}{x}$

$\implies \frac{1}{\sqrt{3}} = \frac{32}{x}$

$\implies x = 32\sqrt{3}$

$\implies y = 32\sqrt{3}\: [From \:(1) ]$

$=32\times 1.732\\= 55.424\:m$

$\therefore Height \:of \:the \:Tower (H)= CE\\= y + 32 \\= 55.424 + 32 \\= 87.424 \:m$

Therefore.,

$\red {Height \:of \:the \:Tower}\green { = 87.424 \:m}$

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