1. A vertical pole 24ft. high is to be braced securely by four wires attached 8ft. from the top. If the wires are fastened to stake in the level ground at the vertices of square with its center at the foot of the pole and sides 20ft. What should be the length of each wire?
2. Work problem 1, if the pole is one the side of the hill that slopes at an angle of 30 degree, and the guy wires are fastened to stakes at the vertices of a square 32ft. on a side. Assume two of sides are parallel to line up the slope of the hill.
#1. well, each wire is the hypotenuse of a right triangle, with legs 16 and 10√2
back later for #2.
To find the length of each wire in both problems, we can use the Pythagorean theorem.
1. Problem 1:
Let's consider the square formed by the guy wires with side length 20ft. To find the length of each wire, we need to find the diagonal of this square, which will be the hypotenuse of a right triangle formed by the diagonal and one side of the square.
Using the Pythagorean theorem, we have:
(diagonal)^2 = (side)^2 + (side)^2
Substituting the values, we get:
(diagonal)^2 = 20^2 + 20^2
(diagonal)^2 = 400 + 400
(diagonal)^2 = 800
diagonal = √800 ≈ 28.28ft.
Since the guy wires are attached 8ft. from the top of the pole, the length of each wire will be the diagonal length minus the distance from the top:
Length of each wire = diagonal - distance from the top
Length of each wire = 28.28ft. - 8ft.
Length of each wire ≈ 20.28ft.
Therefore, the length of each wire should be approximately 20.28ft.
2. Problem 2:
In this problem, we need to consider the hill slope and adjust our calculations accordingly.
Since two sides of the square are parallel to the slope of the hill, we can consider the square formed by the guy wires as a right triangle with a base of 32ft. and a height of 20ft.
The hypotenuse of this right triangle represents the length of each wire. Using the Pythagorean theorem again, we have:
(length of each wire)^2 = (base)^2 + (height)^2
Substituting the values, we get:
(length of each wire)^2 = 32^2 + 20^2
(length of each wire)^2 = 1024 + 400
(length of each wire)^2 = 1424
length of each wire = √1424 ≈ 37.73ft.
Therefore, the length of each wire, considering the slope of the hill, should be approximately 37.73ft.