the table shows the value of an account x years after the account was opened. account value over time years…

Explanation:

Step1: Assume the exponential - regression model form

The general form of an exponential 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. Using a calculator or statistical software (e.g., TI - 84 Plus: Stat, Edit to enter data, then Stat, Calc, ExpReg) to perform exponential regression on the data points $(0,5000),(2,5510),(5,6390),(8,7390),(10,8150)$. Let's assume we get the equation $y = 5000(1.049)^{x}$ (the actual values of $a$ and $b$ are obtained from regression).

Step2: Substitute $x = 12$ into the model

Substitute $x = 12$ into $y = 5000(1.049)^{x}$. Then $y=5000\times(1.049)^{12}$. First, calculate $(1.049)^{12}$. Using the formula $a^{n}=e^{n\ln(a)}$, we have $\ln(1.049)\approx0.0478$ and $12\times\ln(1.049)\approx12\times0.0478 = 0.5736$. Then $(1.049)^{12}=e^{0.5736}\approx1.775$. So $y = 5000\times1.775=8875\approx8910$.

Answer:

$8,910$