Chapter 15.2, Problem 67E

To determine

To calculate: How fast is the concentration increasing at a distance of $1$ centimeter from the center $3$ seconds after the nutrient is introduced when suppose that $1$ cubic centimeter of nutrient is placed at the center of circular petri dish filled with water. It might wonder how the nutrient is distributed after a time of $t$ seconds. According to the classical theory of diffusion, the concentration of nutrient (in parts of nutrient per part of water) after time $t$ is given by:

$u\left(r,t\right)=\frac{1}{4\pi Dt}{e}^{-r^2/\left(4Dt\right)}$,

Here, $D$ is the diffusivity, which we will take to be $1$, and $r$ is the distance from the center in centimeters.