the blades of a windmill turn on an axis that is 30 feet from the ground. the blades are 10 feet long and…

the blades of a windmill turn on an axis that is 30 feet from the ground. the blades are 10 feet long and complete 2 rotations every minute.\nwrite a sine model, y = asin(bt) + k, for the height (in feet) of the end of one blade as a function of time t (in seconds). assume the blade is pointing to the right when t = 0 and that the windmill turns counterclockwise at a constant rate.\no y = 30sin(\\frac{\\pi}{15}t)+10\no y = 30sin(\\frac{\\pi}{15}t)+30\no y = 10sin(\\frac{\\pi}{15}t)+10\no y = 10sin(\\frac{\\pi}{15}t)+30

the blades of a windmill turn on an axis that is 30 feet from the ground. the blades are 10 feet long and complete 2 rotations every minute.\nwrite a sine model, y = asin(bt) + k, for the height (in feet) of the end of one blade as a function of time t (in seconds). assume the blade is pointing to the right when t = 0 and that the windmill turns counterclockwise at a constant rate.\no y = 30sin(\\frac{\\pi}{15}t)+10\no y = 30sin(\\frac{\\pi}{15}t)+30\no y = 10sin(\\frac{\\pi}{15}t)+10\no y = 10sin(\\frac{\\pi}{15}t)+30

Answer

Explanation:

Step1: Determine the amplitude (a)

The amplitude is the length of the blade. Since the blade is 10 feet long, (a = 10).

Step2: Calculate the value of (b)

The wind - mill makes 2 rotations per minute. One rotation is (2\pi) radians. In one minute ((t = 60) seconds), the angle of rotation is (2\times2\pi=4\pi) radians. The general formula for the angular frequency (b) in (y = a\sin(bt)+k) is (b=\frac{\text{angular displacement}}{\text{time}}). So (b=\frac{4\pi}{60}=\frac{\pi}{15}).

Step3: Determine the vertical shift (k)

The axis of the wind - mill is 30 feet from the ground. This is the mid - height of the motion of the end of the blade, so (k = 30).

Answer:

(y = 10\sin(\frac{\pi}{15}t)+30)