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. write 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. y = 30sin(π/15 t)+10 y = 30sin(π/15 t)+30 y = 10sin(π/15 t)+10 y = 10sin(π/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. write 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. y = 30sin(π/15 t)+10 y = 30sin(π/15 t)+30 y = 10sin(π/15 t)+10 y = 10sin(π/15 t)+30

Answer

Explanation:

Step1: Find the amplitude $a$

The amplitude is the length of the blade, so $a = 10$.

Step2: Find the vertical - shift $k$

The axis of the wind - mill is 30 feet from the ground, so $k = 30$.

Step3: Find the value of $b$

The blades complete 2 rotations per minute. One rotation is $2\pi$ radians. In one minute ($t = 60$ seconds), the angle $\theta$ swept is $2\times2\pi$ radians. The general formula for the angular speed is $\theta=bt$. So, $4\pi=b\times60$, and $b=\frac{4\pi}{60}=\frac{\pi}{15}$.

Answer:

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