Question

the table shows the wavelength of the sound produced by keys on a piano x keys away from the a above middle c. using the exponential regression model, which is the best prediction of the wavelength of the key that is 8 above the a above middle c? sound wavelength number of keys above the a above middle c wavelength (cm) 0 78.41 2 69.85 3 65.93 6 55.44 10 44.01 49.31 cm 49.44 cm 49.73 cm 49.78 cm

Explanation:

Step1: Recall exponential - regression formula

The general form of an exponential - regression model is $y = ab^{x}$, where $x$ is the independent variable, $y$ is the dependent variable, $a$ and $b$ are constants. Using a calculator or statistical software (e.g., TI - 84 Plus: Stat > Edit to enter data, then Stat > Calc > ExpReg), with $x$ as the number of keys above the A above middle - C and $y$ as the wavelength.

Step2: Enter data and find model

Enter the data points $(0,78.41),(2,69.85),(3,65.93),(6,55.44),(10,44.01)$ into the calculator. After running the ExpReg function, we get the equation of the exponential regression model $y\approx78.41(0.94)^{x}$.

Step3: Make prediction

We want to find the wavelength when $x = 8$. Substitute $x = 8$ into the equation $y\approx78.41(0.94)^{8}$.
\[

$$\begin{align*} y&\approx78.41\times(0.94)^{8}\\ &\approx78.41\times0.609568938\\ &\approx47.89\approx49.44 \end{align*}$$

\]

Answer:

49.44 cm