An antenna is on a hill that makes a 5 degree angle with the horizon. The antenna is placed so that it is perpendicular to the horizon. A guy wire extends from the top of the antenna to a point 50 ft downhill from the base of the antenna. If the antenna is 25 ft tall, what is the length of the guy wire.
As usual, draw a diagram.
Draw a horizontal line from the base of the wire to the base of the tower. (It will lie underground, but give you a right triangle to work with.)
If we label
T = top of tower
W = base of wire
B = base of tower
P = the point below the base of the tower where the horizontal line from B intersects.
Let u = BP
v = WP
u = 50sin5° = 4.36
v = 50cos5° = 49.81
wire length = √(v^2+(25+u)^2)
= √(49.81^2 + 29.36^2)
= 57.82
To find the length of the guy wire, we can use trigonometry. Let's start by visualizing the situation.
First, draw a right-angled triangle, representing the hill, with the antenna perpendicular to the horizon. Label the length of the antenna as 25 ft. The angle between the hill and the horizon is given as 5 degrees.
Now, let's label the length of the guy wire as "g" (what we're trying to find), and the horizontal distance from the base of the antenna to the end of the guy wire as "x."
Since the antenna is perpendicular to the horizon, we have a right triangle. The length of the antenna (25 ft) is the hypotenuse of the triangle, and the height of the triangle represents the height difference between the top of the antenna and the base.
Using trigonometry, we can determine that the height of the triangle is given by: height = 25 * sin(5°).
Next, we need to determine the length between the base of the antenna and the end of the guy wire (x). We are given that it is 50 ft.
Now, we have two sides of a right triangle: the height (25 * sin(5°)) and the base (x = 50 ft). To find the length of the guy wire (g), we can use the Pythagorean theorem, which states that the square of the hypotenuse (g) is equal to the sum of the squares of the other two sides:
g^2 = (25 * sin(5°))^2 + 50^2.
To find g, we take the square root of both sides of the equation:
g = √[(25 * sin(5°))^2 + 50^2].
Finally, we can calculate the actual length of the guy wire by substituting the values into the equation and evaluating it.