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From two points A and B on the opposite sides of a tower, the angles of elevation to the top of the tower are 45° and 30°, respectively. If the height of the tower is 120 m, then find the distance between A and B, corrected to two decimal places.
  • a)
    207.64 m
  • b)
    327.84 m
  • c)
    207.24 m
  • d)
    357.34 m
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
From two points A and B on the opposite sides of a tower, the angles o...
And 30 degrees, respectively. The distance between points A and B is 100 meters.

To find the height of the tower, we can use the tangent function.

Let's denote the height of the tower as h.

From point A, the angle of elevation to the top of the tower is 45 degrees. This means that the tangent of the angle is equal to the opposite side (height of the tower, h) divided by the adjacent side (distance from point A to the tower, x).

So, tan(45 degrees) = h / x

Similarly, from point B, the angle of elevation to the top of the tower is 30 degrees. This means that the tangent of the angle is equal to the opposite side (height of the tower, h) divided by the adjacent side (distance from point B to the tower, 100 - x).

So, tan(30 degrees) = h / (100 - x)

Now we have two equations:

tan(45 degrees) = h / x

tan(30 degrees) = h / (100 - x)

We can solve these equations simultaneously to find the value of h.

First, let's find the value of x by rearranging the first equation:

x = h / tan(45 degrees)

Substituting this into the second equation:

tan(30 degrees) = h / (100 - h / tan(45 degrees))

Simplifying this equation:

tan(30 degrees) = h / (100 - h / 1)

tan(30 degrees) = h / (100 - h)

Now we can solve for h.

Multiply both sides of the equation by (100 - h):

(100 - h) * tan(30 degrees) = h

Expand the left side of the equation:

100 * tan(30 degrees) - h * tan(30 degrees) = h

Rearrange the equation to isolate h:

h + h * tan(30 degrees) = 100 * tan(30 degrees)

Factor out h:

h * (1 + tan(30 degrees)) = 100 * tan(30 degrees)

Divide both sides of the equation by (1 + tan(30 degrees)):

h = (100 * tan(30 degrees)) / (1 + tan(30 degrees))

Using a calculator, we can find that tan(30 degrees) is approximately 0.5774.

Substituting this value into the equation:

h = (100 * 0.5774) / (1 + 0.5774)

h = 57.74 / 1.5774

h ≈ 36.63

Therefore, the height of the tower is approximately 36.63 meters.
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From two points A and B on the opposite sides of a tower, the angles of elevation to the top of the tower are 45° and 30°, respectively. If the height of the tower is 120 m, then find the distance between A and B, corrected to two decimal places.a)207.64 mb)327.84 mc)207.24 md)357.34 mCorrect answer is option 'B'. Can you explain this answer?
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