A radio antenna is stabilized by two guy wires. One guy wire is 100 m in length and is attached to the top of the antenna. The wire makes an angle of 60° with the ground. One end of the second guy wire is attached to the ground at the same point as the first guy wire. The other end is attached to the antenna such that the wire makes an angle of 45° with the ground. Determine an exact expression for the distance between the points where the two guy wires are attached to the antenna.
To solve this problem, we can use trigonometry and the concept of similar triangles.
Let's denote the distance between the points where the guy wires are attached to the antenna as x.
We can first consider the right triangle formed by the first guy wire, the ground, and the distance x. According to the given information, the angle between the ground and the guy wire is 60°. Using trigonometry, we can write:
sin(60°) = x / 100
Simplifying this equation gives:
x = 100 * sin(60°)
x = 100 * (√3 / 2)
x = 50√3
Next, let's consider the right triangle formed by the second guy wire, the ground, and the distance x. According to the given information, the angle between the ground and the guy wire is 45°. Using trigonometry, we can write:
sin(45°) = x / 100
Simplifying this equation gives:
x = 100 * sin(45°)
x = 100 * (√2 / 2)
x = 50√2
Therefore, the exact expression for the distance between the points where the two guy wires are attached to the antenna is 50√3 units.
To solve this problem, we can use trigonometry. Let's call the distance between the points where the two guy wires are attached to the antenna x.
Using the given information, we can create two right triangles. One triangle has a side of length 100 m and an angle of 60°. The other triangle has a side of length x and an angle of 45°.
In the first triangle, the side adjacent to the angle of 60° is x, and the hypotenuse is 100 m. Therefore, we can use the cosine function to determine the value of x:
cos(60°) = adjacent/hypotenuse
x/100 = cos(60°)
Simplifying the expression, we have:
x = 100 * cos(60°)
x = 100 * 1/2
x = 50 m
Hence, the distance between the points where the two guy wires are attached to the antenna is 50 meters.
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Remarks:
What looks like number 7 with a line in the solution is a small Greek letter chi.
Solution simplification:
50 √3 - 50 = 50 (√3 - 1)