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.
We form a rt. triangle:
X = 6 meters = Hor.
Y = Ver. = Dist. from gnd to junction of wire and pole.
Z = Hyp. = Length of the wire.
A = Angle bet. gnd and wire.
B = Angle bet. pole and wire.
A + B = 90 Deg.
A = 3B.
3B + B = 90
B = 22.5 Deg.
A = 3B = 3*22.5 = 67.5 Deg.
Z*cosA = X
Z = X/cosA = 15.7 m.
tanA = Y/X = Y/6
Y = 6*tanA = 6*tan67.5 = 14.5 m.
To solve this problem, we can use trigonometry. Let's break it down step by step:
1. First, let's define the variables:
- Let θ be the angle that the wire makes with the ground.
- Let 3θ be the angle that the wire makes with the pole.
- Let x be the height of the pole.
2. According to the problem, the wire is pegged 6 meters from the foot of the pole. This forms a right-angled triangle with the pole.
3. In the right-angled triangle, the side opposite θ is x, and the side adjacent to θ is 6 meters.
4. We can use the tangent function to relate θ and x: tan(θ) = opposite/adjacent = x/6.
5. According to the problem, the angle that the wire makes with the pole is three times θ. So, we have: 3θ = 3 * θ.
6. Now, let's use the tangent function again to relate 3θ and x: tan(3θ) = opposite/adjacent = x/6.
7. Since tan(3θ) = 3 * tan(θ), we have: 3 * tan(θ) = x/6.
8. Combining the two equations, we can say that: 3 * tan(θ) = tan(3θ).
9. We can use this equation to solve for θ and find the angle that the wire makes with the ground.
10. Once we find the value of θ, we can find the value of x (the height of the pole) by substituting it into either of the original equations we derived.
11. To solve the equation 3 * tan(θ) = tan(3θ), you can use numerical methods or a graphing calculator to find the value of θ. Once you have the value of θ, substitute it back into one of the original equations to find the height x of the pole.
By following these steps, you should be able to solve the problem and determine the height of the pole.