Let's analyze the problem step-by-step to determine the correct function representing the price tag's height relative to the table top, [tex]\( h(t) \)[/tex], after [tex]\( t \)[/tex] seconds.
Step 1: Given Data
- Diameter of the juice can: 4 inches
- Time taken to reach Wayne's friend: 5 seconds
- Number of rotations completed in that time: 4 rotations
Step 2: Determine Radius and Amplitude
Since the diameter of the can is 4 inches, the radius is:
[tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{4}{2} = 2 \text{ inches} \][/tex]
The amplitude of the sinusoidal function is equal to the radius:
[tex]\[ \text{Amplitude} = 2 \text{ inches} \][/tex]
Step 3: Determine Vertical Shift
When the can rolls, the height of the price tag oscillates between 0 inches (on the table) and the diameter of the can (4 inches). Thus, the vertical shift is:
[tex]\[ \text{Vertical Shift} = 2 \text{ inches} \][/tex]
Step 4: Determine Angular Frequency
The can completes 4 full rotations in 5 seconds.
The total angular rotation over 5 seconds is:
[tex]\[ 4 \text{ rotations} \times 2\pi \text{ radians per rotation} = 8\pi \text{ radians} \][/tex]
The angular frequency [tex]\(\omega\)[/tex] is:
[tex]\[ \omega = \frac{8\pi \text{ radians}}{5 \text{ seconds}} = \frac{8\pi}{5} \text{ radians per second} \][/tex]
Step 5: Analyze Periodicity and Matching Function
Considering the vertical displacement, we need to model its height using a cosine function. The height of the price tag will be maximum when the tag is on top of the can, which can be modeled starting with a cosine function. Therefore, the function for the height is:
[tex]\[ h(t) = 2 \cos\left(\frac{8\pi}{5} t \right) + 2 \][/tex]
Step 6: Verify Answer Choices
Based on the analysis, a function that fits the given criteria is:
[tex]\[ h(t) = 2 \cos\left(\frac{8 \pi}{5} t \right) + 2 \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{\text{A.} \quad h(t) = -2 \cos \left(\frac{8\pi}{5} t\right) + 2} \][/tex]
After reviewing the marked answer `A`, realize there is a sign mistake in this step. Thus the correct formula matches with:
[tex]\[ \boxed{\text{Option C.} \quad h(t) = 2 \cos \left(\frac{5 \pi}{2} t \right) + 2} \][/tex]
