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A vertical tower stands on a horizontal plane and is surmounted by a
nvertical flagstaff. From a point on the ground $30$ m away from the tower,
nwires are attached to the top and bottom of the flagstaff making angles of
nelevation $60^{\circ}$ and $30^{\circ}$ respectively. Find the height of the tower and
nlengths of the wires attached. (Take $\sqrt{3} = 1.73$)
nvertical flagstaff. From a point on the ground $30$ m away from the tower,
nwires are attached to the top and bottom of the flagstaff making angles of
nelevation $60^{\circ}$ and $30^{\circ}$ respectively. Find the height of the tower and
nlengths of the wires attached. (Take $\sqrt{3} = 1.73$)
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Correct figure (1 Mark)
Let $AB$ be the tower of height '$h$' m and $BC$ be the flagstaff.
In right angled $\triangle BAD$
$\frac{h}{30} = \tan 30^{\circ} = \frac{1}{\sqrt{3}}$ (1 Mark)
$\Rightarrow h = \frac{30}{\sqrt{3}} = \frac{30\sqrt{3}}{3} = 10\sqrt{3} = 17.3$ (1/2 Mark)
$\therefore$ height of the tower is $17.3$ m
Also, $\frac{30}{l_1} = \cos 30^{\circ} = \frac{\sqrt{3}}{2}$ (1 Mark)
$\Rightarrow l_1 = \frac{60}{\sqrt{3}} = 20\sqrt{3} = 34.6$ (1/2 Mark)
$\therefore$ length of wire attached to the bottom of the flagstaff is $34.6$ m
In right angled $\triangle CAD$
$\frac{30}{l_2} = \cos 60^{\circ} = \frac{1}{2}$ (1/2 Mark)
$\Rightarrow l_2 = 60$ (1/2 Mark)
$\therefore$ length of wire attached to the top of the flagstaff is $60$ m
Let $AB$ be the tower of height '$h$' m and $BC$ be the flagstaff.
In right angled $\triangle BAD$
$\frac{h}{30} = \tan 30^{\circ} = \frac{1}{\sqrt{3}}$ (1 Mark)
$\Rightarrow h = \frac{30}{\sqrt{3}} = \frac{30\sqrt{3}}{3} = 10\sqrt{3} = 17.3$ (1/2 Mark)
$\therefore$ height of the tower is $17.3$ m
Also, $\frac{30}{l_1} = \cos 30^{\circ} = \frac{\sqrt{3}}{2}$ (1 Mark)
$\Rightarrow l_1 = \frac{60}{\sqrt{3}} = 20\sqrt{3} = 34.6$ (1/2 Mark)
$\therefore$ length of wire attached to the bottom of the flagstaff is $34.6$ m
In right angled $\triangle CAD$
$\frac{30}{l_2} = \cos 60^{\circ} = \frac{1}{2}$ (1/2 Mark)
$\Rightarrow l_2 = 60$ (1/2 Mark)
$\therefore$ length of wire attached to the top of the flagstaff is $60$ m