Applications of Sinusoidal Functions
2) A frog boards a water wheel on the right side from surface of the water. The wheel is half submerged, has a radius of 2 meters, rotates counterclockwise at a constant speed, and completes one full rotation every 4 minutes.
a) Graph two rotations of the frog on this wheel.

Step 1 ：First, we need to find the circumference of the water wheel. Since the radius is \(2\) meters, the circumference is \(4\pi\) meters.

Step 2 ：Next, we need to determine the angular speed of the wheel. Since it completes one full rotation every \(4\) minutes, its angular speed is \(\frac{2\pi}{4}\) radians per minute, or \(\frac{\pi}{2}\) radians per minute.

Step 3 ：Now, we need to find the position of the frog on the wheel as a function of time. Since the wheel rotates counterclockwise, the frog's height above the water surface can be represented by a sinusoidal function. Let \(h(t)\) be the height of the frog above the water surface at time \(t\) minutes. Then, \(h(t) = 2 + 2\sin{\left(\frac{\pi}{2}t\right)}\).

Step 4 ：To graph two rotations of the frog on the wheel, we need to plot the function \(h(t)\) for \(0 \leq t \leq 8\) minutes. The graph will show the height of the frog above the water surface as a function of time, with a period of \(4\) minutes and an amplitude of \(2\) meters.

Step 5 ：\(\boxed{\text{Graph the function } h(t) = 2 + 2\sin{\left(\frac{\pi}{2}t\right)} \text{ for } 0 \leq t \leq 8 \text{ minutes.}}\)