Question

The angle of depression of the top and the bottom
of a 9 m high building from the top of a tower are
30° and 60° respectively. Find the height of the
tower
(a) 8.8 m
(b) 8 m
(c) 7.8 m
(d) 7m​

Answer 1

Correct question:

The angles of depression of the top and the bottom

of a 9 m high building from the top of a tower are

30° and 60° respectively. Find the height of the

tower .

(a) 11 m

(b) 14.3 m

(c) 13.5 m

(d) 7 m​

Given: The angles of depression of the top and bottom of a 9 m high building are 30° and 60°.

To find: The height of the tower.

Answer:

(Diagram for reference attached below.)

In Δ ABC,

$\tt tan\ {60}^{\circ}\ =\ \dfrac{AB}{BC}\\\\\\\bigg[tan\ {60}^{\circ}\ =\ \sqrt{3}\bigg]\\\\\\\sqrt{3}\ =\ \dfrac{x\ +\ 9}{BC}\\\\\\BC\ =\ \dfrac{x\ +\ 9}{\sqrt{3}}\\\\\\BC\ =\ \dfrac{(x\ +\ 9)\sqrt{3}}{3}$

In Δ ADE,

$\tt tan\ {30}^{\circ}\ =\ \dfrac{AE}{DE}\\\\\\\bigg[tan\ {30}^{\circ}\ =\ \dfrac{1}{\sqrt{3}}\bigg]\\\\\\\dfrac{1}{\sqrt{3}}\ =\ \dfrac{x}{\dfrac{\sqrt{3}(x\ +\ 9)}{3}}\\\\\\\dfrac{1}{\sqrt{3}}\ =\ \dfrac{3x}{\sqrt{3}(x\ +\ 9)}\\\\\\\dfrac{\sqrt{3}(x\ +\ 9)}{\sqrt{3}}\ =\ 3x\\\\\\x\ +\ 9\ =\ 3x\\\\\\9\ =\ 3x\ -\ x\\\\\\9\ =\ 2x\\\\\\\dfrac{9}{2}\ =\ x\\\\\\4.5\ =\ x$

Height of the tower, AB = x + 9 = 4.5 + 9 = 13.5 m ⇒ Option (c).