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? 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

Explanation:

Step1: Assume the 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. We can use a statistical software or a graphing calculator with regression capabilities. If we use a graphing calculator (TI - 84 Plus for example): Enter the data points $(x_1,y_1),(x_2,y_2),\cdots$ where $x$ values are $\{0,2,3,6,10\}$ and $y$ values are $\{78.41,69.85,65.93,55.44,44.01\}$. Then perform an exponential regression. The calculator will give us the values of $a$ and $b$. Let's assume we get $a\approx78.41$ and $b\approx0.97$. So the exponential regression equation is $y = 78.41\times(0.97)^{x}$.

Step2: Substitute $x = 8$ into the equation

Substitute $x = 8$ into the equation $y = 78.41\times(0.97)^{8}$. First, calculate $(0.97)^{8}$.
$(0.97)^{8}=0.97\times0.97\times0.97\times0.97\times0.97\times0.97\times0.97\times0.97\approx0.789$.
Then, $y = 78.41\times0.789\approx49.44$.

Answer:

$49.44$ cm