Differential Equations For Math Biology

jimi1987

Seeking Help or a finger in the right direction

Question1

Some cells are growing in a petri dish. At time t = 0 seconds there is one cell, which divides into two cells after one second. Subsequently, every cell divides into two new cells after one second of existence. At t = 60 seconds the dish is full. At what time is the dish half full?


Q2: 2. Suppose the rate of change of a population is proportional to population size.

(a) Suppose the population has 106 members at t = 0 and 107 members after 100 minutes. Find the population size at the end of 150 minutes.

(b) Suppose the population has 106 members at t = 5, and 105 members at t = 15. What was the population at t = 0?

LeixlipRed

The first question is really, really simple. Think about it, if at 60 seconds it's full, go backwards and see what occurs to you.

The second question:

The rate of change of the population is proportional to population size? What does that mean, how could you write that down mathematically. Time we call t, give population size a name, x possibly. That sentence can be written as a differential equation.

jimi1987

thanks for the reply.

the first one is simple but its using the right formula im worried about.

second question ill have a go at thanks

jimi1987

iv used dn/dt (150) = x(10^7)(1.5)(150)

=15000000

wud this be on the right direction?

LeixlipRed

There's no need to use a formula, if you can explain in English what the answer is that's perfectly ok. You can tell us what you think the answer is if you want.

For the second one, you just seem to have thrown all the numbers in. What are you letting n stand for? The general idea in a population model question like this is to form a differential equation (this is ordinary differential equations, you should have one independent variable here, that's t), solve it for the general solution (this will have a constant in it) and then use another "fact" to get a particular solution (i.e. solve for your constant). You need to form the initial DE first. I really think the best thing for you would be to go to the Maths Support Centre, I can see you have difficulty with the fundamentals.

jimi1987

i got the formula from my notes but there not very helpful at all.

i will go tomorrow then , the faster the help the better


thanks

LeixlipRed

Cool, it's open between 3 and 8 tomorrow. If you go in the evening it'll be fairly quiet and you can get proper help. During the afternoon it'll be quite busy.

bnt

jimi1987 wrote: »

thanks for the reply.

the first one is simple but its using the right formula im worried about.

I agree with LeixlipRed - Q1 is a kind of trick question, which you could answer in plain English with no formulas. If you're sure they want a formula, just write down the population at t=60, half it, and see what it looks like. You have to know how Powers work and how to manipulate them.

(n^x)/n = n^(x-1)

Q2: I'd start by putting the statement "the rate of change of a population is proportional to population size" in maths terms: dn/dt ∝ n or dn/dt = k n, where k is some proportionality constant. This is a standard differential equation. (My first though was just "integrate it", but that doesn't provide a solution.)

LeixlipRed

Yeh, that's exactly it. I'd avoid using n though, it normally represents an integer or a natural number and I think this lad is confused enough already!

jimi1987

cheers for the help but i understand it now after help