Question

the accompanying table shows the value of a car over time that was purchased for 13,200 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 11 years.
years (x) 0 1 2 3 4 5 6
value in dollars (y) 13200 11625 10446 9309 8717 7513 6851
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Explanation:

Step1: Recall exponential regression formula

The general form of an exponential regression equation is $y = ab^{x}$, where $a$ and $b$ are coefficients. Using a statistical - calculator or software (since manual calculation of exponential regression for multiple data points is complex), input the data points $(x,y)$ where $x$ is the number of years and $y$ is the value of the car.

Step2: Find the coefficients

After running the exponential regression on the data set $\{(0,13200),(1,11625),(2,10446),(3,9309),(4,8717),(5,7513),(6,6851)\}$, we get the equation $y\approx13200\times(0.897)^{x}$ (rounded to the nearest thousandth for $b$).

Step3: Predict the value at $x = 11$

Substitute $x = 11$ into the equation $y=13200\times(0.897)^{11}$.
$y=13200\times(0.897)^{11}\approx13200\times0.2977\approx3929.64$.

Answer:

$3929.64$