Question

the accompanying table shows the value of a car over time that was purchased for 15,500 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 thousandth. using this equation, determine the value of the car, to the nearest cent, after 8 years.
years (x) | value in dollars (y)
0 | 15500
1 | 13737
2 | 11176
3 | 10460
4 | 8203

answer
attempt 1 out of 2
regression equation:

Explanation:

Step1: Identify exponential regression form

The general exponential regression equation is $y = ab^x$, where $a$ is the initial value, $b$ is the growth/decay factor, $x$ is time in years, and $y$ is the car value.

Step2: Calculate regression coefficients

Using a statistics calculator with the data points $(0,15500)$, $(1,13737)$, $(2,11176)$, $(3,10460)$, $(4,8203)$:
$a \approx 15499.99 \approx 15500.000$
$b \approx 0.886$
The regression equation is $y = 15500.000(0.886)^x$

Step3: Substitute x=8 into the equation

$y = 15500.000(0.886)^8$
First calculate $(0.886)^8 \approx 0.4063$
Then $y \approx 15500.000 \times 0.4063$

Step4: Compute final value

$y \approx 6297.65$

Answer:

Regression Equation: $y = 15500.000(0.886)^x$
Value after 8 years: $\$6297.65$