SAT Question d8ace155: Nonlinear Functions answer and explanation with Desmos solution

Choice A is correct. Since it's given that the account balance, $A\left(t\right)$, in dollars, after $t$ years can be modeled by an exponential function, it follows that function $A$ can be written in the form $A\left(t\right)=N{r}^t$, where $N$ is the initial value of the function and $r$ is a constant related to the growth of the function. It's given that the initial balance of the account is $$36,100.00$, so it follows that the initial value of the function, or $N$, must be $36,100.00$. Substituting $36,100.00$ for $N$ in the equation $A\left(t\right)=N{r}^t$ yields $A\left(t\right)=36,100.00r^t$. It's given that the account balance after $13$ years, or when $t=13$, is $$68,071.93$. It follows that $A\left(13\right)=68,071.93$, or $36,100.00r^{13}=68,071.93$. Dividing each side of the equation $36,100.00r^{13}=68,071.93$ by $36,100.00$ yields $r^{13}=\frac{68,071.93}{36,100.00}$. Taking the $13$th root of both sides of this equation yields $r=\sqrt[13]{\frac{68,071.93}{36,100.00}}$, or $r$ is approximately equal to $1.05$. Substituting $1.05$ for $r$ in the equation $A\left(t\right)=36,100.00r^t$ yields $A\left(t\right)=36,100.00{\left(1.05\right)}^t$, so the equation $A\left(t\right)=36,100.00{\left(1.05\right)}^t$ could define $A$.