SAT Question dc77e0dc: Nonlinear Functions answer and explanation with Desmos solution

Choice A is correct. It's given that the function $f$ models the intensity of an X-ray beam, in number of particles in the X-ray beam, $t$ millimeters below the surface of a sample of iron. When the X-ray beam is at the surface of the sample of iron, it is $0$ millimeters below the surface, so the value of $t$ is $0$. Substituting $0$ for $t$ in the function $f\left(t\right)=500{\left(0.5\right)}^{\frac{t}{12}}$ yields $f\left(0\right)=500{\left(0.5\right)}^{\frac{0}{12}}$. Since any positive number raised to the power of $0$ is equal to $1$, it follows that $f\left(0\right)=500\left(1\right)$, or $f\left(0\right)=500$. Therefore, the estimated number of particles in the X-ray beam at the surface of the sample of iron is $500$.