A vertical pole 35m high, standing on sloping ground is braced by a wire which extends from the top of the pole to a point on the ground 25 m from the foot of the pole. If the pole subtends an angle of 30 degrees at the point where the wire reaches the ground, how long is the wire?
My bad - switched x and y:
(y/x + 1/√3)/(1 - y/x√3) = (y+35)/x
x^2+y^2 = 25^2
x = 19.408, y=15.759
so, z = 54.343
not very far off, really, eh?
it s laborious using a right triangle theorem, however if sine and cosine law for oblique triangle were discussed by the instructor, it s an easy
work.
To find the length of the wire, we can use trigonometry. Let's break down the problem step by step and use the information provided.
Step 1: Draw a diagram.
- Draw a vertical pole that is 35m high, standing on sloping ground.
- Draw a wire that extends from the top of the pole to a point on the ground 25m from the foot of the pole.
- Label the height of the pole as "35m," the distance from the foot of the pole to the point where the wire reaches the ground as "25m," and the angle the pole subtends at the ground as "30 degrees."
Step 2: Identify the right triangle.
- The wire, pole, and the distance from the foot of the pole to the point where the wire reaches the ground form a right triangle.
Step 3: Identify the trigonometric ratio to solve for the length of the wire.
- In this case, we need to find the length of the hypotenuse (the wire). The opposite side is the height of the pole, and the adjacent side is the distance from the foot of the pole to the point where the wire reaches the ground.
Since we are given the opposite side and adjacent side of the triangle, we can use the tangent function to find the length of the wire.
Step 4: Apply the tangent function.
- The tangent of an angle (in this case, the angle subtended at the point where the wire reaches the ground) is equal to the ratio of the opposite side to the adjacent side.
For the given angle of 30 degrees:
tan(30) = opposite/adjacent
tan(30) = 35/25
Step 5: Solve for the length of the wire.
- Multiply both sides of the equation by 25 to isolate the length of the wire:
25 * tan(30) = 35
Using a scientific calculator or trigonometric table, find the tangent of 30 degrees (approximately 0.577):
25 * 0.577 = 14.425
Therefore, the length of the wire is approximately 14.425 meters.
as usual, draw a diagram. Let
T be the top of the pole
B be the bottom of the pole
P be the point where the wire meets the ground.
Draw a horizontal line from P, and extend TB into the ground, so that
Q is the intersection of the horizontal from P and the vertical from T
Then if we let
x be the horizontal distance PQ
y be the vertical distance QB
z be the length of the wire PT
θ be the angle the road makes with the horizontal
we have
y/x = tanθ
(y+35)/x = tan(θ+30°)
So,
tan(θ+30°) = (tanθ + 1/√3)/(1 - tanθ/√3)
(y/x + 1/√3)/(1 - y/x√3) = (x+35)/y
x^2+y^2 = 25^2
That gives us
x = 18.44 and y=16.88
z^2 = x^2 + (y+35)^2
z^2 = 18.44^2 + (16.88+35)^2
z = 55.06
So, the wire is about 55m long.