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?\nsound wavelength\n| number of keys above the a above middle c | wavelength (cm) |\n| ---- | ---- |\n| 0 | 78.41 |\n| 2 | 69.85 |\n| 3 | 65.93 |\n| 6 | 55.44 |\n| 10 | 44.01 |\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?\nsound wavelength\n| number of keys above the a above middle c | wavelength (cm) |\n| ---- | ---- |\n| 0 | 78.41 |\n| 2 | 69.85 |\n| 3 | 65.93 |\n| 6 | 55.44 |\n| 10 | 44.01 |\n49.31 cm\n49.44 cm\n49.73 cm\n49.78 cm

Answer

Explanation:

Step1: Recall exponential - regression formula

The general form of an exponential - regression model is $y = ab^{x}$, where $x$ is the independent variable, $y$ is the dependent variable, $a$ and $b$ are constants. We can use a calculator or statistical software with an exponential - regression function. Let $x$ be the number of keys above the A above middle C and $y$ be the wavelength.

Step2: Input data into calculator/software

Input the data points $(0,78.41),(2,69.85),(3,65.93),(6,55.44),(10,44.01)$ into a calculator or statistical software with an exponential - regression feature. After running the exponential - regression analysis, we get the equation of the regression model.

Step3: Substitute $x = 8$ into the model

Suppose the exponential - regression equation is $y=ab^{x}$. After getting the values of $a$ and $b$ from the regression analysis, substitute $x = 8$ into the equation. Let's assume we use a calculator to perform the exponential regression on the data points. The exponential regression equation for the given data (using a TI - 84 Plus for example) is approximately $y = 78.409\times(0.947)^{x}$. When $x = 8$, we have $y=78.409\times(0.947)^{8}$. First, calculate $(0.947)^{8}\approx0.634$. Then, $y = 78.409\times0.634\approx49.73$.

Answer:

49.73 cm