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

To model the height \( h \) of the end of the windmill blade, we need to account for the rotation of the blade and its starting position. At \( t=0 \), the blade is horizontal and to the right, so we want to use the cosine function. The height oscillates 10 feet above and below the midpoint, which is 35 feet above the ground. Given that the windmill completes a rotation every minute (or 60 seconds), its angular frequency is \( \frac{2\pi}{60} = \frac{\pi}{30} \). Taking these factors into consideration, the correct model would be one that starts from its highest point at \( t=0 \). The height equation that matches these criteria is \( h = 10 \cos\left(\frac{\pi}{30} t\right) + 35 \), but that option isn't listed. Among the options provided, the closest to being correct (but slightly off in terms of angle because it's sin, not cos) is \( h = -10 \sin\left(\frac{\pi}{15} t\right) + 35 \) as it fits the format needed for height oscillation. Now, let’s talk about how this scenario can be seen in real life: windmills are not just for decoration! They’re essential for converting wind energy into mechanical energy. For example, modern wind turbines rely on similar principles, allowing us to harness renewable energy effectively, powering homes, and reducing reliance on fossil fuels. Isn’t it fascinating how what flies overhead can power our everyday lives? Additionally, in problem-solving, a common mistake is not correctly identifying the starting position of the sine or cosine function. Understanding the phase shift and orientation of your oscillating function can save you lots of headaches. Remember, drawing a diagram can help visualize the problem and clarify where to start - don’t skip this step!