SAT Question 752055d1: Nonlinear Functions answer and explanation with Desmos solution

Choice B is correct. It's given that the formula $P=C{\left(2\right)}^{rt}$ gives the number of bacteria in a growth medium, where $r$ and $C$ are constants and $P$ is the number of bacteria $t$ hours after the initial measurement. It's also given that a scientist initially measures $12,000$ bacteria in the growth medium. Since the initial measurement is $0$ hours after the initial measurement, it follows that when $t=0$, $P=12,000$. Substituting $0$ for $t$ and $12,000$ for $P$ in the given equation yields $12,000=C{\left(2\right)}^{r\left(0\right)}$, or $12,000=C{\left(2\right)}^0$, which is equivalent to $12,000=C$. It's given that $4$ hours later, the scientist measures $24,000$ bacteria, or when $t=4$, $P=24,000$. Substituting $4$ for $t$, $24,000$ for $P$, and $12,000$ for $C$ in the given equation yields $24,000=12,000{\left(2\right)}^{4r}$. Dividing each side of this equation by $12,000$ yields $2=2^{4r}$, or $2^1=2^{4r}$, which is equivalent to $1=4r$. Dividing both sides of this equation by $4$ yields $\frac{1}{4}=r$. Therefore, the value of $r$ is $\frac{1}{4}$.