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To predict the wavelength of the key that is 8 keys above the A above middle C using the provided data and an exponential regression model, we'll outline the steps and derive the best prediction.
### Step-by-Step Solution:
1. Data Points:
The given data points are as follows:
- When [tex]\(x = 0\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 78.41\)[/tex] cm.
- When [tex]\(x = 2\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 69.85\)[/tex] cm.
- When [tex]\(x = 3\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 65.93\)[/tex] cm.
- When [tex]\(x = 6\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 55.44\)[/tex] cm.
- When [tex]\(x = 10\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 44.01\)[/tex] cm.
2. Model Selection:
We use an exponential regression model to fit this data. The model is defined as:
[tex]\[ y = a \cdot e^{bx} \][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are parameters to be determined.
3. Determine Parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
The parameters are obtained by fitting the model to the data points provided. After performing the regression analysis, we obtain:
- [tex]\(a \approx 78.40604440735338\)[/tex]
- [tex]\(b \approx -0.05775765170916405\)[/tex]
4. Prediction for 8 keys above A above middle C:
Using the exponential model [tex]\(y = a \cdot e^{bx}\)[/tex], we can predict the wavelength for [tex]\(x = 8\)[/tex]:
[tex]\[ y_{\text{pred}} = a \cdot e^{b \cdot 8} \][/tex]
Substituting the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ y_{\text{pred}} \approx 78.40604440735338 \cdot e^{-0.05775765170916405 \cdot 8} \][/tex]
[tex]\[ y_{\text{pred}} \approx 49.39453584958399 \, \text{cm} \][/tex]
5. Compare Predictions:
We compare the predicted wavelength with the provided choices:
- 49.31 cm
- 49.44 cm
- 49.73 cm
- 49.78 cm
We use the absolute difference to find the closest match:
[tex]\[ |49.39453584958399 - 49.31| \approx 0.0845 \][/tex]
[tex]\[ |49.39453584958399 - 49.44| \approx 0.0455 \][/tex]
[tex]\[ |49.39453584958399 - 49.73| \approx 0.3355 \][/tex]
[tex]\[ |49.39453584958399 - 49.78| \approx 0.3855 \][/tex]
6. Selection of the Best Prediction:
The closest value to our predicted wavelength of 49.39453584958399 cm is 49.44 cm.
### Conclusion:
Therefore, the best prediction of the wavelength of the key that is 8 above the A above middle C is:
[tex]\[ \boxed{49.44 \, \text{cm}} \][/tex]
### Step-by-Step Solution:
1. Data Points:
The given data points are as follows:
- When [tex]\(x = 0\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 78.41\)[/tex] cm.
- When [tex]\(x = 2\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 69.85\)[/tex] cm.
- When [tex]\(x = 3\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 65.93\)[/tex] cm.
- When [tex]\(x = 6\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 55.44\)[/tex] cm.
- When [tex]\(x = 10\)[/tex] keys above A above middle C, the wavelength [tex]\(y = 44.01\)[/tex] cm.
2. Model Selection:
We use an exponential regression model to fit this data. The model is defined as:
[tex]\[ y = a \cdot e^{bx} \][/tex]
where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are parameters to be determined.
3. Determine Parameters [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
The parameters are obtained by fitting the model to the data points provided. After performing the regression analysis, we obtain:
- [tex]\(a \approx 78.40604440735338\)[/tex]
- [tex]\(b \approx -0.05775765170916405\)[/tex]
4. Prediction for 8 keys above A above middle C:
Using the exponential model [tex]\(y = a \cdot e^{bx}\)[/tex], we can predict the wavelength for [tex]\(x = 8\)[/tex]:
[tex]\[ y_{\text{pred}} = a \cdot e^{b \cdot 8} \][/tex]
Substituting the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ y_{\text{pred}} \approx 78.40604440735338 \cdot e^{-0.05775765170916405 \cdot 8} \][/tex]
[tex]\[ y_{\text{pred}} \approx 49.39453584958399 \, \text{cm} \][/tex]
5. Compare Predictions:
We compare the predicted wavelength with the provided choices:
- 49.31 cm
- 49.44 cm
- 49.73 cm
- 49.78 cm
We use the absolute difference to find the closest match:
[tex]\[ |49.39453584958399 - 49.31| \approx 0.0845 \][/tex]
[tex]\[ |49.39453584958399 - 49.44| \approx 0.0455 \][/tex]
[tex]\[ |49.39453584958399 - 49.73| \approx 0.3355 \][/tex]
[tex]\[ |49.39453584958399 - 49.78| \approx 0.3855 \][/tex]
6. Selection of the Best Prediction:
The closest value to our predicted wavelength of 49.39453584958399 cm is 49.44 cm.
### Conclusion:
Therefore, the best prediction of the wavelength of the key that is 8 above the A above middle C is:
[tex]\[ \boxed{49.44 \, \text{cm}} \][/tex]
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