An object attached to a spring oscillates 12 centimeters above and b elow its equilibrium position of 25 centimeters above the table. A stopwatch was started at t = 0 to time the oscillations. The object reached its max height at t = 0.2 seconds. The object reached its lowest height 0.3 seconds after reaching it's max height.

A) Sketch a graph of the function showing one complete cycle, label the max and min valuies and the midlione. Scale the X axis to indicate the period.

B) Formulate the equation of a function of the form h(t) = a cos b(t - c) + d that represents the objects height above the table, h(t) in reltion to time t. Use the function to determine the height of the oject 3.4 secondsa fter thestopwatch started.

Let's work through the given problem step-by-step to construct the graph and find the appropriate function.

### Part A: Sketching the Graph

1. **Understanding the Situation**:
   - The equilibrium position of the object is 25 cm above the table.
   - The object oscillates 12 cm *above* and *below* this equilibrium position, so:
     - **Maximum Height**: \(25 + 12 = 37\) cm.
     - **Minimum Height**: \(25 - 12 = 13\) cm.
   - The **midline** (equilibrium position) is at 25 cm.

2. **Time Analysis**:
   - At \( t = 0 \), the object is at its **maximum height** of 37 cm.
   - It reaches its **minimum height** 0.3 seconds after reaching the maximum.
   - This means it took 0.3 seconds to travel from the maximum to the minimum, i.e., half of a complete cycle.
   - Therefore, a complete cycle (period) is twice this time, or \( 2 \times 0.3 = 0.6 \) seconds.

3. **Key Points** for the Graph:
   - Maximum: At \( t = 0 \), height is 37 cm.
   - Minimum: At \( t = 0.3 \), height is 13 cm.
   - One complete cycle ends at \( t = 0.6 \), bringing the height back to the maximum of 37 cm.

4. **Graph Components**:
   - **X-Axis (Time)**: Label it with a scale from \( t = 0 \) to \( t = 0.6 \) seconds (or beyond for additional cycles).
   - **Y-Axis (Height)**: Mark the maximum, minimum, and midline (13, 25, and 37 cm).
   - Plot the sinusoidal wave showing a complete cycle.

### Part B: Formulating the Equation

We want a function of the form:
\[
h(t) = a \cos(b(t - c)) + d
\]
where:
- \( a \) = Amplitude.
- \( b \) = Frequency factor.
- \( c \) = Horizontal shift.
- \( d \) = Vertical shift.

1. **Identify the Constants**:
   - **Amplitude** \( a \): This is the distance from the midline to the maximum (or minimum). \( a = 12 \).
   - **Vertical Shift** \( d \): This is the midline of the oscillation, which is 25 cm. So, \( d = 25 \).
   - **Frequency Factor** \( b \): The period of the cosine function is given by \( \frac{2\pi}{b} \). We know the period is 0.6 seconds, so:
     \[
     \frac{2\pi}{b} = 0.6 \implies b = \frac{2\pi}{0.6} = \frac{10\pi}{3}
     \]
   - **Horizontal Shift** \( c \): Since the object starts at the maximum height at \( t = 0 \), there is no phase shift, i.e., \( c = 0 \).

2. **Constructing the Function**:
   Putting it all together:
   \[
   h(t) = 12 \cos \left( \frac{10\pi}{3} t \right) + 25
   \]

3. **Finding the Height at \( t = 3.4 \) Seconds**:
   Substitute \( t = 3.4 \) into the function:
   \[
   h(3.4) = 12 \cos \left( \frac{10\pi}{3} \times 3.4 \right) + 25
   \]

I will calculate this value to determine the exact height.The height of the object 3.4 seconds after the stopwatch started is approximately **19 cm**.

### Summary:

1. **Equation**:  
   \[
   h(t) = 12 \cos \left( \frac{10\pi}{3} t \right) + 25
   \]

2. **Height at \( t = 3.4 \) seconds**:  
   \( h(3.4) \approx 19 \) cm.

Would you like to see a detailed breakdown of any step, or do you have any specific questions?

### Related Questions:
1. What would be the object's height at \( t = 1 \) second?
2. How would the graph change if the amplitude increased to 15 cm?
3. What is the object's velocity at \( t = 0.3 \) seconds?
4. How would the period change if the frequency factor \( b \) was doubled?
5. What is the displacement function for this oscillating object?

**Tip**: When analyzing oscillatory motion, always identify the amplitude, midline, and period first for better clarity in constructing equations.

An object attached to a spring oscillates 12 centimeters above and below its equilibrium position of 25 centimeters above the table. A stopwatch was started at t = 0 to time the oscillations. The object reached its max height at t = 0.2 seconds. The object reached its lowest height 0.3 seconds after reaching its max height. A) Sketch a graph of the function showing one complete cycle, label the max and min values and the midline. Scale the X axis to indicate the period. B) Formulate the equation of a function of the form h(t) = a cos b(t - c) + d that represents the object's height above the table, h(t) in relation to time t. Use the function to determine the height of the object 3.4 seconds after the stopwatch started.

Learn how to model the oscillation of an object attached to a spring with a trigonometric function. This problem involves harmonic motion, where the object oscillates 12 cm above and below an equilibrium position of 25 cm. Detailed steps include formulating the equation h(t) = 12 cos((10π/3)t) + 25 and finding the height at t = 3.4 seconds.

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