A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 58°. A guy wire is to be attached to the top of the tower and to the ground, 150 m downhill from the base of the tower. The angle α in the figure is determined to be 7°. Find the length of cable required for the guy wire. (Round your answer to the nearest meter.)
All I know is that this question has something to do with law of sine and that's all.
Thank you!
Sorry I accidentally submitted my question here.
To solve this problem, we can use the Law of Sines. The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of the opposite angle is constant:
a/sin(A) = b/sin(B) = c/sin(C)
In this case, we want to find the length of the guy wire (side c). We know the length of side a (150 m), the angle A (58°), and the angle B (7°).
Let's set up the equation using the Law of Sines:
c/sin(C) = a/sin(A)
Since we know the value of angle C (180° - A - B), we can substitute the known values into the equation:
c/sin(180° - A - B) = 150/sin(58°)
Simplifying the equation, we have:
c/sin(115°) = 150/sin(58°)
Now, we can cross multiply to solve for c:
c * sin(58°) = 150 * sin(115°)
Dividing both sides by sin(58°):
c = (150 * sin(115°)) / sin(58°)
Using a scientific calculator or mathematical software, we can calculate sin(115°) ≈ 0.9063 and sin(58°) ≈ 0.8480.
Substituting these values into the equation, we have:
c ≈ (150 * 0.9063) / 0.8480
Thus, the length of cable required for the guy wire is approximately:
c ≈ 160.3427 meters
Rounding to the nearest meter, the length of the guy wire is approximately 160 meters.
To find the length of the cable required for the guy wire, we can use the Law of Sines.
The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
In this case, we have a triangle with angles of 58°, (180° - α) and α, and sides opposite angles of 150m, the height of the tower, and the length of the guy wire. Let's call the length of the guy wire "x".
Using the Law of Sines, we can set up the following equation:
sin(58°) / 150 = sin(α) / x
Now we can solve for x.
First, let's find sin(α):
sin(α) = sin(7°)
We can now substitute the values back into the equation:
sin(58°) / 150 = sin(7°) / x
To solve for x, we can cross-multiply:
sin(58°) * x = sin(7°) * 150
Now, divide both sides by sin(58°):
x = (sin(7°) * 150) / sin(58°)
Using a calculator, we can evaluate this expression to find the length of the guy wire:
x ≈ 18.511 meters
Therefore, the length of the cable required for the guy wire is approximately 19 meters when rounded to the nearest meter.
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Find the angular speed of the socket.
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