A hydro pole is stabilized at its top by two guy wires of equal length, each of which makes an angle of 60o with the ground. The wires are secured to the ground at points that are 10 m apart and on opposite sides of the pole. What should the length of each wire be and how tall is the hydro pole? Express your answers
pole height: 5/h = cos60°
wire length: √(5^2 + h^2)
why must both wires be secured 5m from the pole?
To solve this problem, we can use trigonometry. Let's label the length of each guy wire as "x" and the height of the hydro pole as "h".
We know that the angle between each guy wire and the ground is 60 degrees. This creates an equilateral triangle between the two guy wires and the ground. Therefore, the angle between each guy wire and the hydro pole is also 60 degrees.
Using trigonometry, we can find the relationship between the height of the pole (h) and the length of each guy wire (x). We can use the sine function, which relates the opposite side of an angle to the hypotenuse.
The sine of 60 degrees is equal to the opposite side (h) divided by the hypotenuse (x).
sin(60) = h / x
Using the value of sin(60) (which is √3 / 2), we can rearrange the equation to find x in terms of h:
√3 / 2 = h / x
Solving for x, we get:
x = 2h / √3
Now, we need to find the length of each guy wire. The distance between the two points on the ground where the guy wires are secured is given as 10 m.
Since the two points are 10 m apart, the distance from each point to the pole (the adjacent side of the angle) is half of that, which is 5 m.
Using trigonometry again, we can find the relationship between x and the adjacent side of the angle. In this case, we can use the cosine function.
The cosine of 60 degrees is equal to the adjacent side (5 m) divided by the hypotenuse (x).
cos(60) = 5 / x
Using the value of cos(60) (which is 1/2), we can rearrange the equation to find x:
1/2 = 5 / x
Solving for x, we get:
x = 10 m
Now that we have found the value of x, we can substitute it back into the equation for x in terms of h:
10 = 2h / √3
To solve for h, we can rearrange the equation:
h = (10 * √3) / 2
Simplifying further, we get:
h = 5√3
Therefore, the length of each guy wire should be 10 m, and the height of the hydro pole is 5√3 meters.