the table shows the wavelength of the sound produced by keys on a piano x keys away from the a above middle…

Explanation:

Step1: Recall exponential regression formula

The general form of an exponential regression model is $y = ab^{x}$, where $x$ is the independent - variable (number of keys above the A above middle C), $y$ is the dependent - variable (wavelength), $a$ and $b$ are constants. We can use a calculator or statistical software with exponential regression capabilities. Let's assume we use a calculator. Enter the data points $(x_1,y_1),(x_2,y_2),\cdots$ where $x$ values are $0,2,3,6,10$ and $y$ values are $78.41,69.85,65.93,55.44,44.01$.

Step2: Obtain the regression equation

After performing exponential regression on the data, we get an equation of the form $y = ab^{x}$. Let's assume the equation is $y = 78.41\times(0.94)^{x}$ (the actual values of $a$ and $b$ are obtained from the regression calculation).

Step3: Substitute $x = 8$

We want to find the wavelength when $x = 8$. Substitute $x = 8$ into the equation $y=78.41\times(0.94)^{8}$. [ \begin{align*} y&=78.41\times(0.94)^{8}\ &=78.41\times0.609568938\ &\approx49.73 \end{align*} ]

Answer:

49.73 cm