SAT Question 7e5a3640: Nonlinear Functions answer and explanation with Desmos solution

Choice D is correct. Let $y$ represent the number of cells per milliliter $x$ hours after the initial observation. Since the number of cells per milliliter doubles every $3$ hours, the relationship between $x$ and $y$ can be represented by an exponential equation of the form $y=a{\left(b\right)}^{\frac{x}{k}}$, where $a$ is the number of cells per milliliter during the initial observation and the number of cells per milliliter increases by a factor of $b$ every $k$ hours. It's given that there were $300,000$ cells per milliliter during the initial observation. Therefore, $a=300,000$. It's also given that the number of cells per milliliter doubles, or increases by a factor of $2$, every $3$ hours. Therefore, $b=2$ and $k=3$. Substituting $300,000$ for $a$, $2$ for $b$, and $3$ for $k$ in the equation $y=a{\left(b\right)}^{\frac{x}{k}}$ yields $y=300,000{\left(2\right)}^{\frac{x}{3}}$. The number of cells per milliliter there will be $15$ hours after the initial observation is the value of $y$ in this equation when $x=15$. Substituting $15$ for $x$ in the equation $y=300,000{\left(2\right)}^{\frac{x}{3}}$ yields $y=300,000{\left(2\right)}^{\frac{15}{3}}$, or $y=300,000{\left(2\right)}^5$. This is equivalent to $y=300,000\left(32\right)$, or $y=9,600,000$. Therefore, $15$ hours after the initial observation, there will be $9,600,000$ cells per milliliter.