Question

the accompanying table shows the value of a car over time that was purchased for 13,900 dollars, where x is years and y is the value of the car in dollars. write an exponential regression equation for this set of data, rounding all coefficients to the nearest hundredth. using this equation, determine the value of the car, to the nearest cent, after 8 years.

answer: attempt 2 out of 2

regression equation: ( y = 13900( )^x )

final answer: 8351

Explanation:

Step1: Use a calculator for exponential regression

Using a statistics calculator (inputting the \(x\) values \(0,1,2,3,4\) and \(y\) values \(13900, 12756, 11254, 9006, 8496\)), the general form of an exponential regression equation is \(y = ab^{x}\).

Step2: Find the coefficients

After running the exponential regression on the data, we get \(a\approx13900\) (since when \(x = 0\), \(y=a\)) and \(b\approx0.89\). So the regression equation is \(y=13900(0.89)^{x}\).

Step3: Calculate the value at \(x = 8\)

Substitute \(x = 8\) into the equation \(y=13900(0.89)^{8}\).

$$y=13900\times(0.89)^{8}$$

$$y = 13900\times0.4304588129$$

$$y\approx6083.38$$

Answer:

The exponential regression equation is \(y = 13900(0.89)^{x}\) and the value of the car after 8 years is approximately \(\$6083.38\)