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? 49.31 cm 49.44 cm 49.73 cm 49.78 cm

Explanation:

Step1: Enter data into calculator

Enter the \(x\) - values (number of keys: \(0,2,3,6,10\)) and \(y\) - values (wavelength: \(78.41,69.85,65.93,55.44,44.01\)) into a graphing calculator for exponential regression.

Step2: Find the regression equation

The general form of an exponential regression equation is \(y = ab^{x}\). After performing the regression on the calculator, we get \(y = 78.33(0.94)^{x}\) (approximate values from regression).

Step3: Substitute \(x = 8\) into the equation

When \(x = 8\), we calculate \(y=78.33\times(0.94)^{8}\).
First, calculate \((0.94)^{8}\approx0.635\).
Then, \(y = 78.33\times0.635\approx49.73\).

Answer:

\(49.73\) cm