Question

A bacteria culture starts with 880 bacteria and grows at a rate proportional to its size. after 5 hours there will be 4400 bacteria. (a) express the population after t hours as a function of t. population: (function of t) (b) what will be the population after 2 hours? (c) how long will it take for the population to reach 2550 ?

Answer 1

Don't know whether or not you've encountered differential equations yet, but will try that approach here.

The growth rate is dy/dt  = ky (which states that the rate is proportional to the size of the population, y, and k is a constant.

Grouping like terms,  

dy
--- = kt, so y = Ne^kt
 y

We are told that at t=0, there are 880 bacteria.  Thus, 880=N.  Therefore,

y = 880e^(kt).  After 5 hours the pop will be 4400; using this info, find k:

4400=880e^(5k), or 5 = e^(5k).    So, our y = 880e^(kt) becomes

                                                                   y = 880e^(5t).

What will be the pop after 2 hours?           y(2)=880e^(10) = 880(22026) =
                                                                   approx. 19,383,290 bacteria

Time to reach a pop of 2550?  2550 = 880e^(5t).  Find t.

ln 2550 = ln 880 + 5t, so ln 2550 - ln 880 = 5t.  Divide both sides by 5 to obtain this time, t.