To model the position of the weight attached to the spring that is oscillating, we can use the sine function in the format:
\[ y(t) = A \sin(Bt) + D \]
where:
- \( A \) is the amplitude,
- \( B \) is related to the period of the oscillation,
- \( D \) is the vertical shift (which can be 0 if we assume the oscillation is centered around the horizontal axis).
Step 1: Determine the Amplitude
The distance from the highest to the lowest point is 4 inches, which means the amplitude \( A \) is half of this distance:
\[ A = \frac{4}{2} = 2 \text{ inches} \]
Step 2: Determine the Period
We are given the period \( T = 3 \) seconds. The relationship between the period and \( B \) is given by:
\[ B = \frac{2\pi}{T} \]
Substituting the value of the period:
\[ B = \frac{2\pi}{3} \]
Step 3: Write the Equation
Considering no vertical shift (assuming the motion is centered around the horizontal axis), the equation becomes:
\[ y(t) = 2 \sin\left(\frac{2\pi}{3} t\right) \]
Looking at the provided responses, we see that:
- The equation \( y = 2 \sin\left(\frac{2\pi}{3} t\right) \) matches our derivation.
Thus, the correct model for the position of the weight at time \( t \) seconds is:
\[ \text{Response: } y = 2 \sin\left(\frac{2\pi}{3} t\right) \]
2 answers