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