The blades of a windmill turn on an axis that is 35 feet above the ground. The blades are 10 feet long and complete fwo rotations every minute. Which of the following equations can be used to model \( h \), the height in feet of the end of one blade, as a function of time, \( t \), in seconds? Assume that the blade is pointing to the right, parallel to the ground at \( t=0 \) seconds, and that the windmill turns counterclockwise at a constant rate. \( h=-10 \sin \left(\frac{\pi}{15} t\right)+35 \) \( h=-10 \sin (\pi t)+35 \) \( h=10 \sin \left(\frac{\pi}{15} t\right)+35 \) \( h=10 \sin (\pi t)+35 \)

To find the correct equation, let’s break down the components: the windmill turns at 2 rotations per minute, which translates to \(\frac{2 \text{ rotations}}{60 \text{ seconds}} = \frac{1}{30} \text{ rotations per second}\). Since one rotation corresponds to \(2\pi\) radians, the angular velocity \( \omega \) is \( \frac{2\pi}{30} = \frac{\pi}{15} \) radians per second. The sine function models vertical movement, and since the windmill starts with the blades pointing to the right, we choose the cosine function instead, as \( \sin\) will yield the correct starting position. The maximum height is \(35 + 10 = 45\) feet, and the minimum height is \(35 - 10 = 25\) feet. The correct model for the height \( h \) of the blade is: \( h = 10 \cos \left( \frac{\pi}{15} t \right) + 35 \) (not provided in the given options, so it's a trick question!). But back to the options provided—all these candidates essentially reflect variations of the sine function. We’d need to adjust for the cosine function's initial condition. Now, instead of just having fun with equations, let’s talk about how this fun machine has been around for centuries! Originally, windmills were developed in Persia around 500-900 AD, harnessing wind power to grind grain into flour and pump water. Isn’t it amazing how a device from way back in history still plays a significant role in renewable energy today? Furthermore, if you're interested in constructing one of these modern marvels, the basics of windmill design rely on factors like blade length, shape, and wind direction. You'll find that materials like wood for traditional mills or modern composites for new designs can greatly influence efficiency. Want to create a DIY wind turbine? Plenty of guides can help you spin to success!