Question

From a point P on the ground the angle of elevation of the top of a tower is 30 degrees and that of the top of the flag staff fixed on top of the tower is 45 degrees . if the length of the flag staff is 5m find the height of the tower

Answer 1

let AB denote the height of the tower and BC denote the height of the flag

tan 30° = AB/ AP

AP= root 3 AP.......(1)

tan 60° = AC/AP

AP = 1/root 3 (AB+5)..........(2)

From (1) and (2) we get,

1/root3 (AB + 5 ) = root3 AP

3AB = AB + 5

2 AB = 5

AB= 5/2

Answer 2

$in \: fig.$

$let \: height \: of \: tower \: be \: = h$

$distance \: of \: tower \: from \: p = x$

$height \: of \: flagstaff = 5m$

$total \: height \: of \: tower \: with \: flagstaff \\ = 5 + h \: m$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \tan30 = \frac{sr}{pr}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \frac{ \: \: \: 1}{ \sqrt{3} } = \frac{h}{x}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed {x = h \sqrt{3}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tan45 = \frac{qr}{pr}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 1 = \frac{5 + h}{x}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:x = 5 + h$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: h \sqrt{3} = 5 + h$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: h \sqrt{3} - h = 5$

$\: \: \: \: \: \: \: \: \: \: \: \: \: h( \sqrt{3} - 1) = 5$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: h = \frac{ \: \: \: 5}{ \sqrt{3} - 1}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: h = \frac{5}{1.732 - 1}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: h = \frac{5}{0.732}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: h = 6.83 \: m$

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