Question

the table shows the value of an account x years after the account was opened. account value over time years after opening account account value 0 $5,000 2 $5,510 5 $6,390 8 $7,390 10 $8,150 based on the exponential regression model, which is the best estimate of the value of the account 12 years after it was opened? $8,910 $8,980 $13,660 $16,040

Explanation:

Step1: Assume the exponential - regression model formula

The general form of an exponential - regression model is $y = ab^{x}$, where $y$ is the account value, $x$ is the number of years after opening the account, $a$ is the initial value, and $b$ is the growth factor. When $x = 0$, $y=a$. From the table, when $x = 0$, $y = 5000$, so $a = 5000$.

Step2: Use another data - point to find $b$

Let's use the point $(2,5510)$. Substitute $a = 5000$, $x = 2$, and $y = 5510$ into $y=ab^{x}$. We get $5510 = 5000b^{2}$. Then $b^{2}=\frac{5510}{5000}=1.102$, and $b=\sqrt{1.102}\approx1.05$.

Step3: Find the value of the account at $x = 12$

Substitute $a = 5000$, $b\approx1.05$, and $x = 12$ into $y = ab^{x}$. So $y=5000\times(1.05)^{12}$. Calculate $(1.05)^{12}\approx1.795856$. Then $y = 5000\times1.795856=8979.28\approx8980$.

Answer:

$8980$ (corresponding to the option $\$8,980$)