Question

A weight is attached to a spring that is oscillating up and down. It takes 3 sec. for the spring to complete one cycle, and the distance from the highest to lowest point is 4 in. What equation models the position of the weight at time t seconds?
(1 point)
Responses

y equals 2 sine left parenthesis Start Fraction pi over 3 End Fraction t right parenthesis
Image with alt text: y equals 2 sine left parenthesis Start Fraction pi over 3 End Fraction t right parenthesis

y equals 4 sine left parenthesis Start Fraction pi over 3 End Fraction t right parenthesis
Image with alt text: y equals 4 sine left parenthesis Start Fraction pi over 3 End Fraction t right parenthesis

y equals 2 sine left parenthesis Start Fraction 2 pi over 3 End Fraction t right parenthesis
Image with alt text: y equals 2 sine left parenthesis Start Fraction 2 pi over 3 End Fraction t right parenthesis

y equals 4 sine left parenthesis Start Fraction 2 pi over 3 End Fraction t right parenthesis
Image with alt text: y equals 4 sine left parenthesis Start Fraction 2 pi over 3 End Fraction t right parenthesis

y equals 2 sine left-parenthesis 3 pi t right-parenthesis
Image with alt text: y equals 2 sine left-parenthesis 3 pi t right-parenthesis

y equals 4 sine left-parenthesis 3 pi t right-parenthesis
Image with alt text: y equals 4 sine left-parenthesis 3 pi t right-parenthesis
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Answer 1

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:

\[
\text{Response: } y = 2 \sin\left(\frac{2\pi}{3} t\right)
\]