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, 50 m downhill from the base of the tower. The angle α in the figure is determined to be 9°. Find the length of cable required for the guy wire. (Round your answer to the nearest meter.)

without the figure, I can't be certain, but I am vislualizing a hill with a tower, the hill slopes downl, and at 50m slant distance from the tower, with a cable guy at an angle of 9deg to the slope.

Now on language, the angle of inclination of the hill (measured to the horizontal) is 58, so the angle of depression is 58, so the angle beween the tower and the hill is 90+58 deg

So I see two angles, and included side, finding the guy wire side

Now notice the angle of the guy to the tower at the top, 180-9-148=23 check that

law of sines
Length/sin148=50/sin23

solve for length

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To find the length of the guy wire, we can use trigonometry and the given angles and distances.

Let's label the relevant points in the figure as follows:
- A: Top of the tower
- B: Base of the tower
- C: Point where the guy wire is attached to the ground (50 m downhill from the base)
- D: Intersection point of the guy wire and the ground

We need to find the length of CD, which is the guy wire.

Since we know the angle of inclination of the hill (58°) and the distance BC (50 m), we can find the length of AB using the sine function:

sin(58°) = AB/BC

AB = BC * sin(58°)
AB = 50 * sin(58°)

Next, we can find the length of BD using the cosine function:

cos(58°) = BD/BC

BD = BC * cos(58°)
BD = 50 * cos(58°)

Now, we need to find the length of CD. We can use the angle α (9°) and the lengths of AB and BD:

tan(α) = AB/BD
CD = AB / tan(α)

CD = (50 * sin(58°)) / tan(9°)

Calculating this expression will give us the length of the guy wire, rounded to the nearest meter.

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