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A vertical tower stands on a horizontal place and is surmounted by a vertical flag staff. At a point on the plane 70 meters away from the tower, an observer notices that the angles of elevation of the top and bottom of the flag-staff are respectively 60° and 45°.

Find the height of the flag staff and that of the tower.

Answer:

**Solution:**

A common notion in trigonometry, specifically, is the angle of elevation, which has to do with height and distance. It is described as an angle formed by the horizontal plane and an oblique line between the observer's eye and a target above it.

Given,

A vertical tower is surmounted by flag staff.

Distance between observer and the tower = 70 m = DC

Angle of elevation of bottom of the flag staff = 45°

Angle of elevation of top of the flag staff = 60°

Let the height of the flag staff = h = AD

Height of tower = H = BC

If we represent the above data in the figure then it forms right angle triangles ΔACD and ΔBCD

When θ is angle in right angle triangle we know that

tan θ = opp. Side/ Adj. side

Now,

tan 45o = BC/ DC

1 = H/ 70

∴ H =70 m

Again,

\begin{aligned} \tan 60^{\circ} &=\frac{A C}{D C} \\ \sqrt{3} &=\frac{A B+B C}{70}=\frac{x+H}{70} \end{aligned}

\begin{aligned} &x+70=70 \sqrt{3} \\ &x=70(\sqrt{3}-1) \end{aligned}

x = 70 (1.732-1)

∴ x = 51.24 m

**Therefore, the height of tower = 70 m and the height of flag staff = 51.24 m**

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