Question

A population of bacteria grows proportional to it's present population at time t. After 3 hours, it is observed that 400 bacteria are present. After 10 hours, there are 2000 present. What was the initial number of bacteria?

Answer 1

we need a model for populations growth, and then a calculator
we can do this one of two ways, easy or hard, but we need a calculator in any case

Answer 2

lets do it the easy way first
reset the clock so that 3 hours is hour zero (when we start counting) and therefore 10 hours, being 7 hours later, is hour 7
we know it goes from 400 to 2000 for a ratio of \(\frac{2000}{400}=5\) meaning the populations is five times as large after 7 hours
we can model this as \(P=400\times 5^{\frac{t}{7}}\)

Answer 3

then you want to know what the population was 3 hours before you began, you can do this by replacing \(t\) by \(-3\) and calculating
\[P(-3)=400\times 5^{\frac{-3}{7}}\] you will need a calculator for the last computation

Answer 4

Okay so I figured out another way now haha! There's three methods now apparently. The way it's been shown in the book does a bunch of stuff with the exponents etc. It's very messy. I seem to have found the same solution by doing something like what you suggested. My solution looks like this: 2000/400=(Po x e^10k)/(Po x e^3k) This gave me 5=e^7k so I solved for k = 0.2299 and put it back into the original equation where 400=Po x e^0.2299(3) I ended up with the same solution in way fewer steps. My only question now is; will this method not work in certain situations? You mentioned an easy and a hard way. Does the easy way only work sometimes?

Answer 5

Thanks a lot for your help too by the way! :D