The angle of elevation of the
top of a vertical tower 50 meters
high from a point X on the ground
is 30 degrees. From a point Y on
the opposite side of the tower, the angle of elevation of the top of
the tower is 60 degrees. Find the
distance between the points X and
Y.
A. 14.43m
B. 77.73m
C. 101.03m
D. 115.47m
Let's assume that the distance between points X and Y is d meters.
From point X, the angle of elevation of the top of the tower is 30 degrees. This means that in a right triangle formed by the tower, the ground, and the line of sight from X to the top of the tower, the opposite side is 50 meters and the adjacent side is d meters. Therefore, we can use the tangent function to find d:
tan(30) = opposite/adjacent
tan(30) = 50/d
Simplifying this equation, we get:
d = 50/tan(30)
Using a calculator, we find that:
d ≈ 86.6025 meters
From point Y, the angle of elevation of the top of the tower is 60 degrees. This time, in the right triangle formed by the tower, the ground, and the line of sight from Y to the top of the tower, the opposite side is 50 meters and the adjacent side is d meters (the same as before). In this case, we can use the tangent function again to find d:
tan(60) = opposite/adjacent
tan(60) = 50/d
Simplifying this equation, we get:
d = 50/tan(60)
Using a calculator, we find that:
d ≈ 28.8675 meters
Since the distance between points X and Y is the sum of the distances from each point to the tower, we can add the two values of d together to find the total distance:
Total distance = 86.6025 + 28.8675 ≈ 115.47 meters
Therefore, the distance between points X and Y is approximately 115.47 meters.
Answer: D. 115.47m
To find the distance between points X and Y, we can use trigonometry.
Let's label the height of the tower as h = 50 meters.
From point X, the angle of elevation to the top of the tower is 30 degrees.
Therefore, we can label the distance from X to the tower as x.
Using trigonometric functions, we can write the following equation:
tan(30 degrees) = h/x
Similarly, from point Y, the angle of elevation to the top of the tower is 60 degrees.
Therefore, we can label the distance from Y to the tower as y.
Using trigonometric functions, we can write the following equation:
tan(60 degrees) = h/y
We now have two equations:
1. tan(30 degrees) = h/x
2. tan(60 degrees) = h/y
By substituting h = 50 meters in both equations, we can solve for x and y.
1. tan(30 degrees) = 50/x
Using a calculator, we find x ≈ 86.6025 meters
2. tan(60 degrees) = 50/y
Using a calculator, we find y ≈ 28.8675 meters
The distance between points X and Y is the sum of x and y:
Distance between X and Y = x + y = 86.6025 + 28.8675 = 115.47 meters
Therefore, the correct answer is D. 115.47m.