A company opens an account with an initial balance of $ 36,100.00 . The account earns

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

Choice B is incorrect. Substituting $0$ for $t$ in this function indies an initial balance of $\$\mathrm{31,971.93}$, rather than $\$\mathrm{36,100.00}$.

Choice C is incorrect. Substituting $0$ for $t$ in this function indies an initial balance of $\$\mathrm{31,971.93}$, rather than $\$\mathrm{36,100.00}$. Additionally, this function indies the account balance is decreasing, rather than increasing, over time. 

Choice D is incorrect. This function indies the account balance is decreasing, rather than increasing, over time.