A guy wire is attached to the top of a radio antenna and to a point on horizontal ground that is 50.0 meters from the base of the antenna. If the wire makes an angle of 59°30' with the ground, approximate the length l of the wire. (Round your answer to the nearest tenth.)
Que lo chupen
as usual, draw a diagram and review your basic trig functions.
L/50.0 = tan 59°30'
To find the length of the guy wire, we can use trigonometry. In this case, we have a right triangle formed by the guy wire, the ground, and a line from the top of the antenna to the ground.
The angle between the guy wire and the ground is given as 59°30' (or 59.5° in decimal form).
We know that the opposite side of the angle is the height of the antenna, and the adjacent side is the distance from the base of the antenna to the point on the ground where the guy wire is attached.
Using the tangent function, we can set up the following equation:
tan(59.5°) = height / 50.0
Solving for the height:
height = 50.0 * tan(59.5°)
Calculating this gives us:
height ≈ 77.8 meters
Finally, we can find the length of the guy wire using the Pythagorean theorem:
l = √(50.0^2 + 77.8^2)
Calculating this gives us:
l ≈ 93.0 meters
Therefore, the approximate length of the guy wire is 93.0 meters (rounded to the nearest tenth).
To approximate the length of the guy wire, we can use trigonometry.
In this case, we have a right triangle formed by the guy wire, the height of the antenna, and the distance on the ground where the wire is attached. The angle between the guy wire and the ground (θ) is given as 59°30'.
To find the length of the guy wire (l), we can use the sine function, which relates the ratio of the length of the side opposite the angle (the height of the antenna) to the length of the hypotenuse (the guy wire).
The sine function is defined as sin(θ) = opposite/hypotenuse. Rearranging the equation, we get:
hypotenuse = opposite / sin(θ)
In this case, the opposite side is the height of the antenna. So, we need to find the sine of 59°30' and then divide the height of the antenna by this value to get the length of the guy wire.
First, convert the angle to decimal degrees:
59°30' = 59 + 30/60 = 59.5 degrees
Then, use a scientific calculator or a trigonometric table to find the sine of this angle.
sin(59.5°) ≈ 0.8571 (rounded to four decimal places)
Now, divide the height of the antenna by the sine of the angle to find the length of the guy wire:
l = height / sin(59.5°)
Plug in the values you have to calculate the length.