To determine the best prediction for the wavelength of the key that is 8 above the A above middle C using an exponential regression model, we can follow these steps:
1. Collect the Data:
The data provided gives us the number of keys above the A above middle C and their corresponding wavelengths.
[tex]\[
\begin{align*}
x &: \{ 0, 2, 3, 6, 10 \} \\
y &: \{ 78.41, 69.85, 65.93, 55.44, 44.01 \} \\
\end{align*}
\][/tex]
2. Model Identification:
We are asked to use an exponential regression model. This means we assume the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be modeled with an equation of the form:
[tex]\[
y = a \cdot b^x
\][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
3. Transform to a Linear Model:
By taking the natural logarithm of both sides of the equation, we can linearize the model:
[tex]\[
\ln(y) = \ln(a \cdot b^x)
\][/tex]
[tex]\[
\ln(y) = \ln(a) + x \cdot \ln(b)
\][/tex]
Letting [tex]\( Y = \ln(y) \)[/tex], [tex]\( A = \ln(a) \)[/tex], and [tex]\( B = \ln(b) \)[/tex], the model becomes:
[tex]\[
Y = A + Bx
\][/tex]
Thus, we perform a linear regression on [tex]\( x \)[/tex] vs. [tex]\( \ln(y) \)[/tex].
4. Find the Coefficients:
Using linear regression on the transformed data:
[tex]\[
x: \{ 0, 2, 3, 6, 10 \}
\][/tex]
[tex]\[
Y: \{ 4.361951470242831, 4.246350085702971, 4.188593573125209, 4.015301354881648, 3.784416880823 \}
\][/tex]
The linear regression yields coefficients that represent [tex]\( \ln(a) \)[/tex] and [tex]\( \ln(b) \)[/tex]:
[tex]\[
\ln(y) = -0.05775194578546567 \cdot x + 4.361880845254086
\][/tex]
Therefore:
[tex]\[
A = 4.361880845254086, \quad B = -0.05775194578546567
\][/tex]
Converting back:
[tex]\[
a = e^{A} \approx 78.40446249017809
\][/tex]
[tex]\[
b = e^{B} \approx 0.9438840528175181
\][/tex]
5. Prediction for 8 Keys Above:
Using the exponential model [tex]\( y = a \cdot b^x \)[/tex]:
[tex]\[
y(8) = 78.40446249017809 \cdot (0.9438840528175181)^8
\][/tex]
This calculation gives:
[tex]\[
y(8) \approx 49.39579400502101 \text{ cm}
\][/tex]
6. Closest Given Option:
We compare this predicted value to the provided options: [tex]\( 49.31 \)[/tex], [tex]\( 49.44 \)[/tex], [tex]\( 49.73 \)[/tex], and [tex]\( 49.78 \)[/tex] cm. The closest value to [tex]\( 49.39579400502101 \)[/tex] cm is [tex]\( 49.44 \)[/tex] cm.
Therefore, the best prediction for the wavelength of the key that is 8 above the A above middle C is:
[tex]\[
\boxed{49.44 \text{ cm}}
\][/tex]
