SAT Question 0a74365c: Nonlinear Functions answer and explanation with Desmos solution

Let $y$ represent the population of Smithville $x$ years after $1659$. Since the population doubled every $75$ years, the relationship between $x$ and $y$ can be represented by an exponential equation in the form $y=a{\left(b\right)}^{\frac{x}{k}}$, where $a$ is the population of Smithville in $1659$ and the population increases by a factor of $b$ every $k$ years. It’s given that the population doubled, or increased by a factor of $2$, every $75$ years. Therefore, $b=2$ and $k=75$. Substituting $2$ for $b$ and $75$ for $k$ in the equation $y=a{\left(b\right)}^{\frac{x}{k}}$ yields $y=a{\left(2\right)}^{\frac{x}{75}}$. It's also given that the population of this town was $240,000$ in $1959$. The number of years from $1659$ to $1959$ is $1959-1659$, or $300$ years. Therefore, when $x=300$, the value of $y$ is $240,000$. Substituting $300$ for $x$ and $240,000$ for $y$ in the equation $y=a{\left(2\right)}^{\frac{x}{75}}$ yields $240,000=a{\left(2\right)}^{\frac{300}{75}}$, or $240,000=a{\left(2\right)}^4$, which is equivalent to $240,000=a(16)$. Dividing both sides of this equation by $16$ yields $15,000=a$. Therefore, the population of this town in $1659$ was $15,000$.