How to model an exponential growth problem?

Question

Let's say I have 5000 bacteria at $t$ (time) $= 0 $, and 8000 bacteria at $t = 10$. Assuming the number of bacteria grow exponentially, how do I model an equation for this situation to help me find the amount of bacteria present at $t=30$?

Answer 1

The exponential function $y(t) = Ae^{kt}$ solves the differential equation $y^\prime = ky$ with initial condition $y(0) = A$. As you are given $y(0)=5000$, let $A=5000$ and get $y(t) = 5000e^{kt}$. Then substitute $y(10) = 8000$ and derive the value of $k$.

Answer 2

The number of bacteria will follow $N=Ae^{at}$ for some constants $A,a$. If you plug your data in, you get two equations in two unknowns. The $t=0$ one is particularly simple, allowing you to solve for $A,a$ easily.