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[Arithela]

1. In a certain bacterial culture, the rate of increase is proportional to the number of bacteria present.
a) If the number doubles in 3 hours, find the hourly growth rate.

 

[Pwnage101]

classic 2U question - it means dP/dt = kP, k positive constant, can find P by getting reciprocal, integrating, then making P the subject

 

[Starcraftmazter]

My solution:
(Note B0 = B subscript 0)

B = B0 * e ^ kt
Given:
2B0 * e ^ kt = B0 * e ^ k(t+3)

2 * e ^ kt = e ^ kt+3k
2 * e ^ kt = ( e ^ kt ) ( e ^ 3k )
2 = e ^ 3k
ln 2 = 3k
k = ( ln 2 ) / 3

Therefore, the hourly growth rate is ( ln 2 ) / 3

 

[Arithela]

ok, so if the question says "IN 3 hours" - we add + 3 to t?
if question says "after 3 hours" - do we sub in t = 3?

is that right?

 

[Arithela]

the answer says dN/dt = 0.23N

so it must have been a different method i tihnk... something with N in it

 

[Starcraftmazter]

(ln 2) / 3 is precisely 0.23104906 - which is 0.23 to 2 decimal places.
N is just whatever unit the growth rate would be measured in.

As for adding / subbing, it depends on the question. Try and think what the question means in terms of the formula - so for example, this question said, "in 3 hours", so this means 3 hours more than the arbitrary example, so we take t to represent the number of hours, and add 3 to t to signify 3 more hours.

 

[Arithela]

that cleared it all up, thanks!

 

[Ennaybur]

for fucks sake. you should not be helping people with maths hw, on a forum, at almost 4am, on a fri night (/sat morning)

 

[Forbidden.]

for fucks sake. you should not be helping people with maths hw, on a forum, at almost 4am, on a fri night (/sat morning)

fuck.

he <3 math

 

[Starcraftmazter]

N is not a unit, remember the growth rate dN/dt is equal to kN

k is not the growth rate otherwise youre implying a constant rate of change.

Mmm, growth constant?

Technical definitions bore me.

 

[Starcraftmazter]

No, I was asking whether k is referred to as a growth constant. Ie. a constant value which determines the growth rate.