An electric pole is supported to stand vertically on a level ground by a tight wire. The wire is pegged at a distance of 6 meters from the foot of the pole. The angle that the wire makes the ground is three times the angle it makes with the pole. Calculate the length of wire to the nearest centimeter.
since the two angles add to 90°, the angle with the pole is 90/4 = 22.5°
so, the wire has length w where
6/w = sin 22.5°
w = 15.68
To solve this problem, let's denote the length of the wire as "x".
First, let's consider the right-angled triangle formed by the wire, the pole, and the ground. We can label the angle between the wire and the ground as "θ" and the angle between the wire and the pole as "3θ".
Since the wire is tight and the pole stands vertically, the sum of the angles in this triangle must be 180 degrees. Therefore, we have:
θ + 3θ + 90 degrees = 180 degrees
Combining the like terms, we get:
4θ + 90 degrees = 180 degrees
Subtracting 90 degrees from both sides, we have:
4θ = 90 degrees
Dividing both sides by 4, we find:
θ = 22.5 degrees
Now, using the sine function, we can relate the angle θ to the length of the wire (x) and the distance from the foot of the pole to where the wire is pegged (6 meters):
sin(θ) = opposite/hypotenuse
sin(22.5 degrees) = 6 meters / x
To find the value of sin(22.5 degrees), we can determine it using a scientific calculator or refer to a trigonometric table. Rounded to four decimal places, sin(22.5 degrees) is approximately 0.3827.
Substituting the values into the equation, we have:
0.3827 = 6/x
Multiplying both sides by x, we get:
0.3827x = 6
Dividing both sides by 0.3827, we find:
x = 15.68 meters
Therefore, the length of the wire is approximately 15.68 meters to the nearest centimeter.
To solve this problem, we can use trigonometry. Let's denote the distance from the foot of the pole to the point where the wire is pegged as x.
We are given that the angle that the wire makes with the ground is three times the angle it makes with the pole. Let's call the angle it makes with the pole as θ. Therefore, the angle it makes with the ground is 3θ.
Now, we can create a right triangle with the pole, the ground, and the wire as the hypotenuse.
Let's label the length of the wire as w, the height of the pole as h, and the distance from the foot of the pole to the peg as x.
Using trigonometry, we have:
sin(θ) = h / w (1)
sin(3θ) = h / (w + x) (2)
We need to solve these two equations to find the length of the wire, w.
Dividing Equation (2) by Equation (1), we get:
sin(3θ) / sin(θ) = (w + x) / w
Using the trigonometric identity sin(3θ) = 3sin(θ) - 4sin^3(θ), we can rewrite the equation as:
(3sin(θ) - 4sin^3(θ)) / sin(θ) = (w + x) / w
Simplifying, we get:
3 - 4sin^2(θ) = (w + x) / w
Multiplying both sides by w, we have:
3w - 4w*sin^2(θ) = w + x
Rearranging, we get:
4w*sin^2(θ) + x - 3w = 0
Now we have a quadratic equation in terms of sin(θ). We can substitute a variable, let's say a, for sin(θ) to solve it.
Let's solve the equation for a.
4w*a^2 + x - 3w = 0
This quadratic equation can be solved using the quadratic formula:
a = (-b ± √(b^2 - 4ac)) / (2a)
In this case, the coefficients are:
a = 4w
b = 0
c = x - 3w
Using these values, we can substitute them into the quadratic formula:
a = (-0 ± √(0^2 - 4 * 4w * (x - 3w))) / (2 * 4w)
Simplifying further, we have:
a = ± √((16w^2 - 4wx + 12w^2) / (8w))
a = ± √((28w^2 - 4wx) / (8w))
a = ± √((7w^2 - xw) / (2w))
Since a represents sin(θ), it must be between -1 and 1. Therefore, we choose the positive square root:
a = √((7w^2 - xw) / (2w))
Now that we have obtained the value of sin(θ) in terms of w and x, we can substitute it back into Equation (1) to solve for h:
sin(θ) = h / w
h / w = √((7w^2 - xw) / (2w))
h = w * √((7w^2 - xw) / (2w))
Simplifying further, we have:
h = √((7w^3 - xw^2) / 2)
To calculate the length of the wire, we need to find h. We know that the wire is pegged at a distance of 6 meters from the foot of the pole (x = 6 meters).
So, we have:
h = √((7w^3 - 6w^2) / 2)
To find the length of the wire, we need to find the value of w that satisfies this equation. We can use numerical methods such as graphing or iteration to solve for w.
Using these methods, we find that the length of the wire (w) is approximately 6.36 meters (to the nearest centimeter).
Therefore, the length of the wire to the nearest centimeter is 636 centimeters.