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Select the correct answer from the drop-down menu. A vertical pole is supported by two ropes staked to the ground on opposite sides of the p
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Select the correct answer from the drop-down menu. A vertical pole is supported by two ropes staked to the ground on opposite sides of the pole. One rope is 8 meters long, and the other is 7 meters. The distance between the stakes is 6 meters, and the height of the pole is meters.
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Mathematics
5 years
2021-08-20T19:46:06+00:00
2021-08-20T19:46:06+00:00 1 Answers
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Answer:
6.78 m
Step-by-step explanation:
Since the set up forms a triangle with length of ropes two sides of the triangle a = 8 m and b = 7 m respectively, the distance between them which is c = 6 m forms the third side of the triangle.
To find the height of the pole, we need to find the angle between any of the ropes and the ground. So, choosing the 8 m long rope side and using the cosine rule,
b² = a² + c² – 2accosФ where Ф is the angle opposite the 7 m long side which is also the angle between the 8 m long side and the ground.
So, making, Ф subject of the formula, we have
b² – (a² + c²) = 2accosФ
cosФ = [b² – (a² + c²)]/2ac
Ф = cos⁻¹{[b² – (a² + c²)]/2ac}
substituting the values of the variables into the equation, we have
Ф = cos⁻¹{[b² – (a² + c²)]/2ac}
Ф = cos⁻¹{[7² – (8² + 6²)]/2(8)(6)}
Ф = cos⁻¹{[49 – (64 + 36)]/96}
Ф = cos⁻¹{[49 – 100]/96}
Ф = cos⁻¹{-51/96}
Ф = cos⁻¹{-0.53125}
Ф = 122.09°
Since the height of the pole, h is a perpendicular bisector to the base of the triangle, and the 8 m long side form a triangle with it and the ground and the 8 m long side being the hypotenuse side of this triangle, we have that
sinФ = h/a where a = 8 m and Ф = 122.09° = the angle between the 8 m long side and the ground.
h = asinФ
substituting the values of the variables into the equation, we have
h = asinФ
h = (8 m)sin122.09°
h = 8 m × 0.8472
h = 6.78 m