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The graph shown represents a sinusoidal function, where the height \( y \) of the sensor above the ground varies over time \( t \) as the bicycle wheel spins. The key points of the graph can help derive the equation of the form \( y(t) = A \sin(B(t - h)) + k \), where:

- \( A \) is the amplitude.
- \( B \) relates to the period.
- \( h \) is the horizontal shift.
- \( k \) is the vertical shift.

### Step-by-step breakdown:

1. **Amplitude** (\( A \)):  
   The amplitude is the distance from the midline to a maximum or minimum point. From the graph, the maximum value is 58 cm and the minimum value is 22 cm.  
   - The midline (average of max and min):  
     \[
     \text{Midline} = \frac{58 + 22}{2} = 40 \, \text{cm}
     \]
   - Amplitude:  
     \[
     A = 58 - 40 = 18 \, \text{cm}
     \]

2. **Period** (\( T \)):  
   The period is the time it takes to complete one full cycle. Based on the points given:  
   - From \( t = 0.75 \) to \( t = 3.75 \), the graph completes half of a cycle (from a peak to the next peak), which corresponds to \( 3 \) seconds. Thus, the full period \( T \) is:
     \[
     T = 6 \, \text{seconds}
     \]
   - The value of \( B \), which relates to the period, is given by:
     \[
     B = \frac{2\pi}{T} = \frac{2\pi}{6} = \frac{\pi}{3}
     \]

3. **Vertical Shift** (\( k \)):  
   The midline of the function is at 40 cm, which represents the vertical shift \( k \). Therefore, \( k = 40 \).

4. **Horizontal Shift** (\( h \)):  
   The graph reaches its first peak at \( t = 0.75 \), so the horizontal shift is \( h = 0.75 \).

### Equation:
Combining all the information, the equation becomes:
\[
y(t) = 18 \sin\left( \frac{\pi}{3}(t - 0.75) \right) + 40
\]

This represents the sensor's height \( y \) above the ground as a function of time \( t \).

Would you like any details or further explanations?  
Here are 5 related questions you might consider:
1. How would the equation change if the maximum height of the sensor increased?
2. What would the equation look like if the period of the spinning wheel changed to 8 seconds?
3. How do the sine and cosine functions differ when modeling this scenario?
4. Can this equation be written using a cosine function instead of sine?
5. How does the horizontal shift affect the timing of the peaks in the graph?

**Tip:** Always pay attention to the period when working with sinusoidal functions; it's key in understanding how the function repeats over time!

Write an equation in terms of y, height in centimeters above the ground, and t, time in seconds, to represent the given context, based on the graph of the bicycle wheel sensor's height.

This page provides a step-by-step solution for finding the sinusoidal equation representing the height of a sensor attached to a bicycle wheel. The problem involves understanding key trigonometric concepts like amplitude, period, and phase shift. Suitable for high school students in Grades 10-12.

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