To determine which function accurately represents the oscillations of the spring, we need to analyze the given requirements and each option step by step:
1. Amplitude: The amplitude is the peak value of the oscillation, given here as 5. This corresponds to the coefficient in front of the sine function.
2. Frequency: The frequency is given by [tex]\(\frac{3}{2 \pi}\)[/tex]. In the general sine function [tex]\( f(t) = A \sin(Bt) + C \)[/tex], the term [tex]\(B\)[/tex] relates to the frequency. Specifically, the frequency is [tex]\( \frac{B}{2\pi} \)[/tex], which means [tex]\( B = 3 \)[/tex] here.
3. Midline: The midline is the vertical shift from the zero baseline, given here as 4. This corresponds to the constant term added or subtracted outside of the sine function.
Given this information, let’s analyze each option:
1. Option 1: [tex]\( f(t) = 5 \sin(4t) + 3 \)[/tex]
- Amplitude: 5 (Correct)
- Frequency: [tex]\(\frac{4}{2\pi} = \frac{2}{\pi}\)[/tex] (Incorrect)
- Midline: 3 (Incorrect)
2. Option 2: [tex]\( f(t) = 3 \sin(5t) + 4 \)[/tex]
- Amplitude: 3 (Incorrect)
- Frequency: [tex]\(\frac{5}{2\pi}\)[/tex] (Incorrect)
- Midline: 4 (Correct)
3. Option 3: [tex]\( f(t) = 4 \sin(3t) + 5 \)[/tex]
- Amplitude: 4 (Incorrect)
- Frequency: [tex]\(\frac{3}{2\pi}\)[/tex] (Correct)
- Midline: 5 (Incorrect)
4. Option 4: [tex]\( f(t) = 5 \sin(3t) + 4 \)[/tex]
- Amplitude: 5 (Correct)
- Frequency: [tex]\(\frac{3}{2\pi}\)[/tex] (Correct)
- Midline: 4 (Correct)
From the detailed analysis, Option 4 is the only function that correctly matches the given amplitude of 5, frequency of [tex]\(\frac{3}{2 \pi}\)[/tex], and midline of 4.
Thus, the correct function representing the oscillations of the spring is:
[tex]\[ f(t) = 5 \sin(3t) + 4 \][/tex]
