Two poles of equal height are standing opposite to each other on either side of a road which is 100 m wide. Find a point between them on road, angles of elevation of their tops are30and60. The height of each pole in meter, is:a)25√3b)20√3c)28√3d)30√3Correct answer is option 'A'. Can you explain this answer?

Question

Two poles of equal height are standing opposite to each other on either side of a road which is 100 m wide. Find a point between them on road, angles of elevation of their tops are30and60. The height of each pole in meter, is:a)25&radic;3b)20&radic;3c)28&radic;3d)30&radic;3Correct answer is option 'A'. Can you explain this answer?

Answer

Answer

Let the height of the poles be

and the distances from the point on the road to the poles be

and

meters. Since the angles of elevation are

and

, we can use trigonometric ratios to find

.

For the pole where the angle of elevation is

:

For the pole where the angle of elevation is

:

We know that

and

. Substituting these values into the equations, we get:

Now, solve for

from both equations:

Equating the two expressions for

:

Solving for

:

Substitute

into

to find the height

:

Therefore, the correct answer is:

Answer

Let's assume the height of each pole is h meters.

We can form two right triangles with the given information:
In the first triangle:
angle of elevation = 30 degrees
opposite side = h
adjacent side = x (distance from the point between the poles to one of the poles)
We can use the tangent function: tan(30) = opposite/adjacent = h/x

In the second triangle:
angle of elevation = 60 degrees
opposite side = h
adjacent side = 100 - x (distance from the point between the poles to the other pole)
We can use the tangent function: tan(60) = opposite/adjacent = h/(100-x)

Now we can solve the two equations simultaneously to find the value of h.

From the first equation, we have:
tan(30) = h/x
1/sqrt(3) = h/x
x = sqrt(3)h

Substituting this value of x in the second equation, we have:
tan(60) = h/(100 - sqrt(3)h)
sqrt(3) = h/(100 - sqrt(3)h)
sqrt(3)(100 - sqrt(3)h) = h
300 - 3h = h^2
h^2 + 3h - 300 = 0

Using the quadratic formula, we find two possible values for h: h = 15 or h = -20. However, since the height cannot be negative, we discard the solution h = -20.

Therefore, the height of each pole is 15 meters.

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