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, \( y=a \sin (b t)+k \), 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. \( y=30 \sin \left(\frac{\pi}{15} t\right)+10 \) \( y=30 \sin \left(\frac{\pi}{15} t\right)+30 \) \( y=10 \sin \left(\frac{\pi}{15} t\right)+10 \) \( y=10 \sin \left(\frac{\pi}{15} t\right)+30 \)

To find the height of the end of one blade of the windmill, let’s first break down the components of the sine model \(y = a \sin(bt) + k\). The amplitude \(a\) is half the distance the blade moves vertically; thus, with a 10-foot blade, \(a = 10\). The vertical shift \(k\) is the height of the axis plus the amplitude, giving us \(k = 30 + 10 = 30\). Lastly, the factor \(b\) relates to how fast the windmill rotates, completing 2 rotations per minute or \(\frac{2\pi}{30}\) radians per second, so \(b = \frac{\pi}{15}\). So the model properly reads: \(y = 10 \sin\left(\frac{\pi}{15} t\right) + 30\). Now about real-world applications: Windmills are not just historical relics; they’re still found across the globe, helping to generate renewable energy. By converting kinetic energy from wind into mechanical or electrical energy, modern wind turbines play a vital role in sustainable practices, powering homes, businesses, and even entire cities. And for some common mistakes: One of the most frequent errors while formulating sine models is misidentifying the midline or the amplitude. It’s crucial to clearly separate the rotational movement’s vertical position from the motion itself. Always ensure the axis height is factored in correctly along with verifying your amplitude—that’s half the height of the total swing and should arise directly from the physical dimensions of the system!