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The angles of elevation to the top of a tower from two points that are on the same side of the tower and 30 meters apart are 37.6 degrees and 46.3 degrees. What is the height of the tower?

A. 15.2 meters
B. 20.7 meters
C. 25.4 meters
D. 30.1 meters

Answer :

Final Answer:

The height of the tower is **(b) 20.7 meters**.

Explanation:

To find the height of the tower, we can use trigonometric principles involving the tangent function. The tangent of the angle of elevation to the top of the tower from a certain point is equal to the height of the tower divided by the horizontal distance from the point to the base of the tower.

Let's denote:

- h as the height of the tower,

- d as the horizontal distance from the points to the base of the tower.

We have two equations based on the given angles of elevation:

1. For the first point: tan(37.6°) = h / d

2. For the second point: tan(46.3°) = h / (d + 30)

Now, we can solve these equations simultaneously to find the height of the tower. First, we rearrange each equation to solve for h:

1. h = d * tan(37.6°)

2. h = (d + 30) * tan(46.3°)

Setting these two expressions equal to each other:

d * tan(37.6°) = (d + 30) * tan(46.3°)

Now, we can solve for d:

d = (30 * tan(46.3°)) / (tan(37.6°) - tan(46.3°))

Once we have the value of d, we can substitute it back into either of the original equations to find the height of the tower. After the calculations, we find that the height of the tower is approximately 20.7 meters.

Therefore, the correct option is **(b) 20.7 meters**.

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