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Answer :
To solve this problem, we need to fit a sinusoidal function of the form [tex]\( y = A \sin(\omega x - \phi) + B \)[/tex] to the given precipitation data. Here's how we can approach it step-by-step:
### Step 1: Understanding the Data
The given data consists of monthly precipitation amounts for a city, measured in inches:
- January (Month 1): 6.06 inches
- February (Month 2): 4.45 inches
- March (Month 3): 4.38 inches
- April (Month 4): 2.08 inches
- May (Month 5): 1.27 inches
- June (Month 6): 0.56 inches
- July (Month 7): 0.17 inches
- August (Month 8): 0.46 inches
- September (Month 9): 0.91 inches
- October (Month 10): 2.24 inches
- November (Month 11): 5.21 inches
- December (Month 12): 5.51 inches
### Step 2: Plotting the Scatter Diagram
First, we plot the scatter diagram to visualize the data:
- Create a graph with `x`-axis representing months (1 to 12) and `y`-axis representing precipitation in inches.
- Plot the individual points for each month with their corresponding precipitation value.
### Step 3: Determine Initial Parameters
To fit the sinusoidal function, we need initial guesses for the parameters [tex]\( A \)[/tex], [tex]\( \omega \)[/tex], [tex]\( \phi \)[/tex], and [tex]\( B \)[/tex]:
- Amplitude [tex]\( A \)[/tex]: Estimate as half of the range of data, [tex]\(\frac{\text{max value} - \text{min value}}{2}\)[/tex].
- Vertical Shift [tex]\( B \)[/tex]: Initial guess can be the average of the precipitation values.
- Phase Shift [tex]\( \phi \)[/tex]: Start initially with 0 or you can adjust based on when the maximum and minimum occur.
- Angular Frequency [tex]\( \omega \)[/tex]: The period is 12 months, so [tex]\( \omega = \frac{2\pi}{12} \)[/tex].
### Step 4: Fitting the Sinusoidal Model
Use a mathematical tool or calculator capable of curve fitting—like a graphing calculator or a software package—to determine the precise values for the parameters by fitting the function to the data.
- Input the initial guesses.
- Allow the tool to optimize these parameters to fit the data best.
### Step 5: Plot the Sinusoidal Function
Once you have the parameters, plot the sinusoidal function on the same graph as your scatter plot:
- Use the fitted values of [tex]\( A \)[/tex], [tex]\( \omega \)[/tex], [tex]\( \phi \)[/tex], and [tex]\( B \)[/tex] to draw the curve.
- Compare the sinusoidal curve to your scatter plot to visually assess the fit.
### Conclusion
By following these steps, you can derive parameters that provide a sinusoidal curve fitting well with the precipitation data. The exact values of [tex]\( A, \omega, \phi, \)[/tex] and [tex]\( B \)[/tex] will depend upon the fitting tool's output, ensuring that the curve aligns closely with seasonal variation in precipitation data. This fitted sinusoidal equation can then be used to predict or model similar periodic patterns in other data sets.
### Step 1: Understanding the Data
The given data consists of monthly precipitation amounts for a city, measured in inches:
- January (Month 1): 6.06 inches
- February (Month 2): 4.45 inches
- March (Month 3): 4.38 inches
- April (Month 4): 2.08 inches
- May (Month 5): 1.27 inches
- June (Month 6): 0.56 inches
- July (Month 7): 0.17 inches
- August (Month 8): 0.46 inches
- September (Month 9): 0.91 inches
- October (Month 10): 2.24 inches
- November (Month 11): 5.21 inches
- December (Month 12): 5.51 inches
### Step 2: Plotting the Scatter Diagram
First, we plot the scatter diagram to visualize the data:
- Create a graph with `x`-axis representing months (1 to 12) and `y`-axis representing precipitation in inches.
- Plot the individual points for each month with their corresponding precipitation value.
### Step 3: Determine Initial Parameters
To fit the sinusoidal function, we need initial guesses for the parameters [tex]\( A \)[/tex], [tex]\( \omega \)[/tex], [tex]\( \phi \)[/tex], and [tex]\( B \)[/tex]:
- Amplitude [tex]\( A \)[/tex]: Estimate as half of the range of data, [tex]\(\frac{\text{max value} - \text{min value}}{2}\)[/tex].
- Vertical Shift [tex]\( B \)[/tex]: Initial guess can be the average of the precipitation values.
- Phase Shift [tex]\( \phi \)[/tex]: Start initially with 0 or you can adjust based on when the maximum and minimum occur.
- Angular Frequency [tex]\( \omega \)[/tex]: The period is 12 months, so [tex]\( \omega = \frac{2\pi}{12} \)[/tex].
### Step 4: Fitting the Sinusoidal Model
Use a mathematical tool or calculator capable of curve fitting—like a graphing calculator or a software package—to determine the precise values for the parameters by fitting the function to the data.
- Input the initial guesses.
- Allow the tool to optimize these parameters to fit the data best.
### Step 5: Plot the Sinusoidal Function
Once you have the parameters, plot the sinusoidal function on the same graph as your scatter plot:
- Use the fitted values of [tex]\( A \)[/tex], [tex]\( \omega \)[/tex], [tex]\( \phi \)[/tex], and [tex]\( B \)[/tex] to draw the curve.
- Compare the sinusoidal curve to your scatter plot to visually assess the fit.
### Conclusion
By following these steps, you can derive parameters that provide a sinusoidal curve fitting well with the precipitation data. The exact values of [tex]\( A, \omega, \phi, \)[/tex] and [tex]\( B \)[/tex] will depend upon the fitting tool's output, ensuring that the curve aligns closely with seasonal variation in precipitation data. This fitted sinusoidal equation can then be used to predict or model similar periodic patterns in other data sets.
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