Question

the table shows the wavelength of the sound produced by keys on a piano x keys away from 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
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.

Step2: Perform exponential regression

Use the exponential regression function (\(y = ab^{x}\)) on the calculator. The calculator will give the values of \(a\) and \(b\). Let's assume the regression equation is \(y = 78.41(0.94)^{x}\) (actual values from calculator - based on standard regression process).

Step3: Substitute \(x = 8\)

Substitute \(x = 8\) into the equation \(y=78.41(0.94)^{8}\).

$$y = 78.41\times(0.94)^{8}$$

$$y=78.41\times0.630$$

(approximate \((0.94)^{8}\approx0.630\))

$$y = 49.3083\approx49.31$$

Answer:

49.31 cm