A wind storm caused a 64 ft pole to lean due easwt at an angle of 63.8 degrees to the ground. A 106 ft long guy wire is then attached to the top of the pole and anchored in the ground due west of the foot of the pole. How far from the foot of the pole is the guy wire anchored?
I see in the triangle the following:
obtuse angle: 180-63.8, opposite side 106ft, another side 64.
Law of sines to get the guy angle:
64/sinA=106/sin(180-63.8)
solve for angle A.
Then, knowing angle A, you can find the third angle at the top between the guywire and the flagpole.
law of sines:
SinA/64=SinThirdAngle/how far
solve for how far.
To find the distance from the foot of the pole to the guy wire's anchor point, we can use trigonometry. Let's break down the problem step by step:
1. Draw a diagram: Draw a triangle to represent the situation. Label the base of the triangle as the distance from the foot of the pole to the guy wire's anchor point. Label the height of the triangle as the length of the pole (64 ft). Label the hypotenuse as the length of the guy wire (106 ft).
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h | .
| .
--------.
| .
| θ.
| .
2. Identify the relevant trigonometric relationship: In this case, we have the opposite side (height) and the hypotenuse, which suggests we can use the sine function:
sin(θ) = opposite / hypotenuse
We need to find the opposite side, which is the distance from the foot of the pole to the guy wire's anchor point and solve for it.
3. Solve for the opposite side: Rearrange the sine function to solve for the opposite side.
opposite = sin(θ) * hypotenuse
= sin(63.8°) * 106 ft
Using a calculator, calculate sin(63.8°), then multiply the result by 106 ft to find the distance from the foot of the pole to the guy wire's anchor point.
4. Calculate the answer:
sin(63.8°) ≈ 0.895 (rounded to three decimal places)
distance = 0.895 * 106 ft
≈ 94.67 ft (rounded to two decimal places)
Therefore, the guy wire is anchored approximately 94.67 ft from the foot of the pole.