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
V.
A. 14.43m
B. 77.73m
C. 101.03m
D. 115.47m
Let's label the points as follows:
X: Point on the ground where the angle of elevation is 30 degrees
Y: Point on the opposite side of the tower where the angle of elevation is 60 degrees
V: Top of the tower
Now, we can create a right triangle with the following sides:
1) Side VY: This is the height of the tower, which is given as 50 meters.
2) Side XY: This is the distance between points X and Y, which is what we need to find.
3) Side XV: This is the distance between point X and the top of the tower, which is what we need to find.
We can use trigonometry to solve this problem. Let's start with finding the distance XY.
In triangle VXY, we can use the tangent function to relate the angle of elevation and the sides of the triangle:
tan(angle) = opposite/adjacent
In triangle VXY, the angle of elevation at point X is 30 degrees. So, we have:
tan(30) = XY/VY
We know that VY is 50 meters. Therefore, we can solve for XY:
XY = VY * tan(30)
XY = 50 * tan(30)
XY ≈ 28.87 meters
Now, let's find the distance XV.
In triangle VXY, we can use the tangent function again to relate the angle of elevation and the sides of the triangle:
tan(angle) = opposite/adjacent
In triangle VXY, the angle of elevation at point Y is 60 degrees. So, we have:
tan(60) = XV/XY
We know that XY is 28.87 meters. Therefore, we can solve for XV:
XV = XY * tan(60)
XV = 28.87 * tan(60)
XV ≈ 49.74 meters
So, the distance between points X and V is approximately 49.74 meters.
Therefore, the correct answer is A. 14.43m.
Let's denote the distance between points X and V as "d".
Using the information given, we can create a right-angled triangle with the tower as the vertical height, point X as the base, and angle of elevation 30 degrees. Let's call this triangle "Triangle X".
Similarly, we can create another right-angled triangle with the tower as the vertical height, point Y as the base, and angle of elevation 60 degrees. Let's call this triangle "Triangle Y".
In Triangle X, the opposite side is the height of the tower (50m) and the angle is 30 degrees. Therefore, we can use the tangent function to find the length of the base (d).
tan(30) = opposite/adjacent
tan(30) = 50/d
To solve for d, we can rearrange the equation:
d = 50 / tan(30)
d ≈ 86.6025404
In Triangle Y, the opposite side is also the height of the tower (50m), and the angle is 60 degrees. Therefore, we can use the tangent function to find the length of the base (d):
tan(60) = opposite/adjacent
tan(60) = 50/(100 - d)
Solving for d, we can rearrange the equation:
d = (100 - d) * tan(60)
d = 100 * tan(60) - d * tan(60)
d + d * tan(60) = 100 * tan(60)
d(1 + tan(60)) = 100 * tan(60)
d(1 + √3) = 100 * √3
d ≈ 100 * √3 / (1 + √3)
Simplifying this, we can multiply the numerator and denominator by (1 - √3):
d ≈ (100 * √3 / (1 + √3)) * (1 - √3) / (1 - √3)
d ≈ (100√3 - 300) / (-2)
d ≈ (-300 + 100√3) / 2
d ≈ -150 + 50√3
Since distance cannot be negative, we take the positive value:
d ≈ 50√3 - 150
So, the distance between points X and V is approximately 50√3 - 150 meters.
To find the exact value, we can use a calculator:
d ≈ 50*(1.732) - 150
d ≈ 86.6025404 - 150
d ≈ -63.3974596
As we can see, this value is negative, which is not possible.
Therefore, the closest option to the distance between points X and V is option C. 101.03 meters.