A guy wire 11.2m long. It supports a vertical power pole. The wire is fastened to the ground 9.87m from the base of a 5.3m tall pole. Calculate the measure of the guy wire and the ground
28 degree
To find the measure of the guy wire and the ground, we can use the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the guy wire, vertical power pole, and the ground form a right triangle.
Let's call the measure of the guy wire "x" and the distance from the base of the pole to the ground "y".
According to the Pythagorean Theorem, we have:
x^2 = y^2 + (11.2 - 9.87)^2 [Substituting the values]
x^2 = y^2 + (1.33)^2
x^2 = y^2 + 1.7689
Now, we need another equation to solve for x and y. Since the guy wire is 11.2m long and is fastened to the ground 9.87m from the base of the pole, we can write:
x + y = 11.2 [Equation 2]
Now, we have a system of two equations:
x^2 = y^2 + 1.7689 [Equation 1]
x + y = 11.2 [Equation 2]
To solve this system, we can substitute the value of y from Equation 2 into Equation 1:
(x + y)^2 = y^2 + 1.7689
Expanding (x + y)^2, we get:
x^2 + 2xy + y^2 = y^2 + 1.7689
Subtracting y^2 from both sides, we have:
x^2 + 2xy = 1.7689
Now, let's substitute the value of x from Equation 2 into this equation:
(11.2 - y)^2 + 2(11.2 - y)y = 1.7689
Simplifying and expanding, we get:
125.44 - 22.4y + y^2 + 22.4y - 2y^2 = 1.7689
Combining like terms and simplifying further:
- y^2 + 0.44 - 1.7689 = 0
Rearranging the equation, we have:
-y^2 - 1.3289 = 0
Multiplying through by -1, we get:
y^2 + 1.3289 = 0
Now, we can solve this quadratic equation by applying the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 1, b = 0, and c = 1.3289.
Using the quadratic formula, we find:
y = (± √(0^2 - 4(1)(1.3289))) / (2(1))
Simplifying further:
y = (± √(0 - 5.3156)) / 2
y = (± √(-5.3156)) / 2
Since √(-5.3156) is an imaginary number, we can conclude that there is no real solution for y in this case.
Therefore, we cannot determine the exact measures of the guy wire and the ground based on the given information.