SAT Question 52b1700c: Nonlinear Functions answer and explanation with Desmos solution

Choice C is correct. It's given that the relationship between $t$ and $n$ is exponential. The table shows that the value of $n$ increases as the value of $t$ increases. Therefore, the relationship between $t$ and $n$ can be represented by an increasing exponential equation of the form $n=a{\left(1+b\right)}^t$, where $a$ and $b$ are positive constants. The table shows that when $t=0$, $n=604$. Substituting $0$ for $t$ and $604$ for $n$ in the equation $n=a{\left(1+b\right)}^t$ yields $604=a{\left(1+b\right)}^0$, which is equivalent to $604=a\left(1\right)$, or $604=a$. Substituting $604$ for $a$ in the equation $n=a{\left(1+b\right)}^t$ yields $n=604{\left(1+b\right)}^t$. The table also shows that when $t=1$, $n=606.42$. Substituting $1$ for $t$ and $606.42$ for $n$ in the equation $n=604{\left(1+b\right)}^t$ yields $606.42=604{\left(1+b\right)}^1$, or $606.42=604\left(1+b\right)$. Dividing both sides of this equation by $604$ yields approximately $1.004=1+b$. Subtracting $1$ from both sides of this equation yields that the value of $b$ is approximately $0.004$. Substituting $0.004$ for $b$ in the equation $n=604{\left(1+b\right)}^t$ yields $n=604{\left(1+0.004\right)}^t$. Therefore, of the choices, choice C best represents the relationship between $t$ and $n$.