A vertical tower stands on level ground. Two…

A vertical tower stands on level ground. Two points A and B lie on a straight line through the foot of the tower, on the same side of it, with B…

Single correct option

A vertical tower stands on level ground. Two points

A

and

B

lie on a straight line through the foot of the tower, on the same side of it, with

B

farther from the tower and

AB=40

m. The angles of elevation of the top of the tower from

A

and

B

are

60^\circ

and

30^\circ

respectively. The height of the tower, in metres, is

Show answer and solution

Answer: (C) $20\sqrt3$

Let the tower have height

h

and let the foot be

O

. From

A

the angle of elevation is

60^\circ

, so

OA=\dfrac{h}{\tan60^\circ}=\dfrac h{\sqrt3}

. From

B

the angle is

30^\circ

, so

OB=\dfrac h{\tan30^\circ}=h\sqrt3

. Since

A

and

B

are on the same side of the foot and

B

is farther,

AB=OB-OA=h\sqrt3-\dfrac h{\sqrt3}=\dfrac{2h}{\sqrt3}

. Setting this equal to

40

gives

h=20\sqrt3

m. Check:

OB=60

OA=20

m, so

AB=40

m, and

\tan30^\circ=\dfrac{20\sqrt3}{60}

. Adding the two distances (as if

A

and

B

were on opposite sides) would give

h=10\sqrt3

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