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: Input data into calculator

Enter the data points \((x,y)\) (where \(x\) is the number of keys and \(y\) is the wavelength) into a graphing calculator or statistical software for exponential regression. The general form of an exponential regression model is \(y = ab^{x}\).

Step2: Find the regression equation

After performing the exponential regression on the data \((0,78.41),(2,69.85),(3,65.93),(6,55.44),(10,44.01)\), we get the equation \(y = 78.41(0.94)^{x}\) (approximate values of \(a = 78.41\) and \(b=0.94\) based on regression calculations).

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

When \(x = 8\), we calculate \(y=78.41\times(0.94)^{8}\).
First, calculate \((0.94)^{8}\approx0.629\).
Then, \(y = 78.41\times0.629\approx49.31\).

Answer:

49.31 cm