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

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?\n\nsound wavelength\nnumber of keys above the a above middle c | wavelength (cm)\n0 | 78.41\n2 | 69.85\n3 | 65.93\n6 | 55.44\n10 | 44.01\n\n49.31 cm\n49.44 cm\n49.73 cm\n49.78 cm

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?\n\nsound wavelength\nnumber of keys above the a above middle c | wavelength (cm)\n0 | 78.41\n2 | 69.85\n3 | 65.93\n6 | 55.44\n10 | 44.01\n\n49.31 cm\n49.44 cm\n49.73 cm\n49.78 cm

Answer

Explanation:

Step1: Assume exponential regression model

The general form of an exponential regression model is $y = ab^{x}$, where $y$ is the wavelength, $x$ is the number of keys above the A above middle - C, $a$ and $b$ are constants. Using a calculator or software (e.g., TI - 84 Plus: Stat, Edit to enter data, then Stat, Calc, ExpReg) with the data points $(0,78.41),(2,69.85),(3,65.93),(6,55.44),(10,44.01)$.

Step2: Obtain regression equation

After running the exponential regression on the data, we get an equation of the form $y\approx78.41\times(0.94)^{x}$.

Step3: Substitute $x = 8$

Substitute $x = 8$ into the equation $y\approx78.41\times(0.94)^{8}$. First, calculate $(0.94)^{8}=0.94\times0.94\times0.94\times0.94\times0.94\times0.94\times0.94\times0.94\approx0.634$. Then, $y\approx78.41\times0.634\approx49.73$.

Answer:

49.73 cm