Question

I was looking through my old tests and found a short but interesting problem. You guys give it a quick try!

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A culture initially has \(N_0\) number of bacteria. At time \(t=1\) hour the number of bacteria is measured to be \((3/2)N_0\). If the rate of growth is proportional to the number of bacteria present, determine the time necessary for the number of bacteria to triple.

Hint: the model to use is\[\frac{dP}{dt}=P.\]

Answer 1

My apologies, the model should be\[\frac{dP}{dt}=kP,\]where \(k\) is a constant. It has to be more general.

Answer 2

exponential growth problem:

P = e^kt
Let N0 = 1
3/2 = e^k
--> k = ln(3/2)

Plug in to find t when P = 3N0
e^kt = 3
(3/2)^t = 3
--> t = ln(3)/ln(3/2) = 2.71

Answer 3

@dumbcow: you got it! :P

Answer 4

http://www2.fiu.edu/~aladrog/ModelingFirstOrderEq.pdf

Answer 5

If the number of bacteria at time t is \(P(t)\), then the rate of change is given by:
\[\frac{dP}{dt}=cP\]
The solution of this Homogeneous DE is \(P(t)=ke^{ct}.\)
Then at time t=0 \(P(0)=k=N_0\) and at t=1,
\(\frac{3}{2}N_0=N_0e^c \implies c=\ln{\frac{3}{2}}.\)

Answer 6

Now, we have the formula we can find thethe number of bacteria at any given time t. Or we can find when it's troubled too.

Answer 7

what kind of math is this?

Answer 8

Ordinary differential equations.

Answer 9

in English please

Answer 10

:0

Answer 11

It's a post-calculus subject. :)

Answer 12

pokemon!! How's it going?

Answer 13

@pokemon23: Something we take after a 1-year of single variable calculus.

Answer 14

hey AnwarA going fine :) I finally understand the concept of electrical circuits

Answer 15

pokemon's head: "What's single-variable calculus?" :P

Answer 16

^ LOL

Answer 17

ok thanks ffm I'm in Algebra hopefully I will go into further into math because I want to be an engineer

Answer 18

I don't how to answer that question

Answer 19

some people take this simultaneously with multi-variable and some even with Linear algebra.

Answer 20

Good luck pokemon! :D

Answer 21

Why are you twisting with electrical circuits when you are taking Algebra ?

Answer 22

I take electricity in hs (freshman)

Answer 23

@FoolForMath: That's right! I was then taking cal 3, and most of my friends were taking linear algebra.

Answer 24

Linear algebra it's very helpful with vec calc

Answer 25

math these days

Answer 26

You want to become an Electrical Engineer? @pokemon

Answer 27

Math is more than helpful. Its even used to figure crimes

Answer 28

Linear algebra is very useful to multi-variable too, lets say we can't understand multivariable calculus without moderate understanding of Linear algebra.

Answer 29

@Katrina like the ones of Numb3rs?

Answer 30

@KatrinaKaif: Numb3rs was a cool show!

Answer 31

well yes anwar also become a pilot in aviation

Answer 32

And a doctor too?

Answer 33

Maybe a lawyer as well? ;P

Answer 34

@across and @katrina: I love Numb3rs too, Markov chain and max-min network flows rules :D

Answer 35

nah I can't be a doctor

Answer 36

I hate looking at blood

Answer 37

Most definitively! Its absurd that such mechanics in math comes useful. They do have mathematicians working on crime!

Answer 38

freaks me out

Answer 39

I finally found Numb3rs fans haha

Answer 40

Haha, just teasing you man! I studied Electrical Engineering for 3 semesters, I can probably help if you want to ask anything! :D @pokemon

Answer 41

Mathematics is great tool, however I don't know how people study pure mathematics, it's the application that make it so AWESOME :D

Answer 42

Hi got a question!

Answer 43

I sorry

Answer 44

across is the best one to answer you FFM!

Answer 45

@AnwarA: Liar.. you told me that you are a pure math student :P

Answer 46

I'd like a criminal investigating job =)

Answer 47

@FoolForMath: I like to think of it this way: Pure mathematicians develop the tools whilst applied mathematicians use them!

Answer 48

I didn't!

Answer 49

Yeah that too applied kat :D

Answer 50

Sheikhs are liars. Processed and Confirmed.

Answer 51

thanks AnwarA I will gladly need your help when I get a midterm in electricity circuits

Answer 52

lol @kat

Answer 53

What did I do to you Katrina? :(

Answer 54

I don't know where the Kirchoff laws come from

Answer 55

I think across would kick us if she can! I'm off to chat! :)

Answer 56

@across: Yes I agree, but what's the fun in developing when you are not applying it ? ;)

Answer 57

well I know they come from Maxwell equations, but i cand not derived it by myself

Answer 58

Lol Anwar =) Just my way of communicating to you

Answer 59

@victorarana: Let's consider something more elementary, I know Fermat's last theorem but I can't prove it :P

Answer 60

@FFM: You get to brag about the usefulness of your tools! No, but I know what you mean. :P I simply love the elegance of it.

Answer 61

I don't want a proof I just want to see the process of derivation.

Answer 62

Well the theorem is elementary but not the proofs :P

Answer 63

Derivatives are more than easy =P

Answer 64

@KatrinaKaif: Well not in the time of Newton ;)

Answer 65

They're easy right now because there were guys who cracked their heads proving\[f(x)=x^n,\]\[f'(x)=nx^{n-1}\](though that one's not hard to derive).

Answer 66

;P

Answer 67

LOL!

Answer 68

@across: 100 years from now people will say Measure theory is easy :P

Answer 69

Will they?

Answer 70

I completely agree with you. lol

Also, calculus will be an elementary school course.