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? 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 49.31 cm 49.44 cm 49.73 cm 49.78 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? 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 49.31 cm 49.44 cm 49.73 cm 49.78 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 statistical software (e.g., TI - 84 Plus: Stat > Edit to enter data, then Stat > Calc > ExpReg), with the data points $(x_1,y_1)=(0,78.41)$, $(x_2,y_2)=(2,69.85)$, $(x_3,y_3)=(3,65.93)$, $(x_4,y_4)=(6,55.44)$, $(x_5,y_5)=(10,44.01)$. After running the exponential regression, we get the equation $y\approx78.41(0.94)^{x}$.

Step2: Substitute $x = 8$

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

Answer:

49.73 cm