To predict the wavelength of the key that is 8 keys above the A above middle C using an exponential regression model, we need to:
1. Determine the relationship between the number of keys and the wavelength.
2. Use this relationship to predict the wavelength for 8 keys above A above middle C.
Given the data points:
| Number of keys above the A above middle C (x) | Wavelength (cm) |
|-----------------------------------------------|-----------------|
| 0 | 78.41 |
| 2 | 69.85 |
| 3 | 65.93 |
| 6 | 55.44 |
| 10 | 44.01 |
1. Model the relationship:
The data suggests an exponential decay in the wavelength as the number of keys increases. The general exponential function can be written as:
[tex]\[
y = a \cdot e^{bx}
\][/tex]
where [tex]\(y\)[/tex] is the wavelength, [tex]\(x\)[/tex] is the number of keys, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are coefficients determined from the data.
2. Identify the coefficients:
By fitting the provided data to an exponential model using regression techniques, we obtain the coefficients:
[tex]\[
a \approx 78.40446249017809
\][/tex]
[tex]\[
b \approx -0.05775194578546567
\][/tex]
3. Predict the wavelength for [tex]\(x = 8\)[/tex]:
Substitute [tex]\(x = 8\)[/tex] into the exponential model:
[tex]\[
y = 78.40446249017809 \cdot e^{-0.05775194578546567 \cdot 8}
\][/tex]
Calculating this yields:
[tex]\[
y \approx 49.39579400502102
\][/tex]
Therefore, the best prediction of the wavelength of the key that is 8 keys above the A above middle C is approximately [tex]\(49.40 \, \text{cm}\)[/tex].
Comparing this with the provided options:
- [tex]$49.31 \, \text{cm}$[/tex]
- [tex]$49.44 \, \text{cm}$[/tex]
- [tex]$49.73 \, \text{cm}$[/tex]
- [tex]$49.78 \, \text{cm}$[/tex]
The closest option is [tex]\(49.44 \, \text{cm}\)[/tex].
Thus, the best prediction of the wavelength is [tex]\(49.44 \, \text{cm}\)[/tex].
