Question

A bacteria culture starts with 800 bacteria and grows at a rate proportional to its size. After 2 hours there will be 1600 bacteria.
(a) Express the population after t hours as a function of t.
population

Answer 1

The population is modeled by \(y=Ce^{kt},\) where C is the initial population, e is the constant 2.718..., k is a relative growth factor, t is time, and y is the population after some time t.

You're given that \(C=800\), and that \(y(2)=1600\) (i.e., \(1600=800e^{2k}\)).
First thing to do is to solve the equation \(1600=800e^{2k}\) for k.

Answer 2

so take the ln of 2 then divide that answer by 2 yes?

Answer 3

the function then would equal 800e^(.3465735903t) right?

Answer 4

Yep, \(k=\frac{\ln2}{2}\), which makes your equation
\[y=800e^{\frac{\ln2}{2}t}\]
(Rounded is also fine.)