Exponential Growth, two variables?

[brianwolk]

The count in a bateria culture was 300 after 20 minutes and 1200 after 40 minutes.

What was the initial size of the culture?

Find the doubling period.

Find the population after 75 minutes.

When will the population reach 10000

starting with the model n(t)=N[sub:1vvzxcnr]0[/sub:1vvzxcnr]e[sup:1vvzxcnr]rt[/sup:1vvzxcnr]

so, i have tried calculating the rate of change between the two given counts with x=time in minutes and y=bacteria count. to take care of the variable r=relative rate of growth expressed as a proportion of the population.

1200-300=45
40-20

so ive tired that and am not sure whether plugging that number into the equation will help but if i do i get

n(t)=N[sub:1vvzxcnr]0[/sub:1vvzxcnr]e[sup:1vvzxcnr]45t[/sup:1vvzxcnr]

and if i plug values from above i get

n(20)=N[sub:1vvzxcnr]0[/sub:1vvzxcnr]e[sup:1vvzxcnr](45)(20)[/sup:1vvzxcnr]=300

and it errors out on me. ive been struggling with this problem for some time now, any help would be appreciated. Thank you so much

 

[galactus]

The count in a bacteria culture was 300 after 20 minutes and 1200 after 40 minutes.

What was the initial size of the culture?

\(\displaystyle N=N_{0}e^{kt}\)

\(\displaystyle 300=N_{0}e^{20k}\)

\(\displaystyle N_{0}=300e^{-20k}\)

Sub into:

\(\displaystyle 1200=N_{0}e^{40k}\)

and get \(\displaystyle 1200=300e^{20k}\)

Solve for k and get \(\displaystyle k=\frac{ln(2)}{10}\approx .0693147\)

That means \(\displaystyle N_{0}=75\)

Now, you can finish the other portions. OK?.

 

[brianwolk]

thank you so much

i have a couple clarifying questions though.
N=N[sub:38r0fai0]0[/sub:38r0fai0]e[sup:38r0fai0]tk[/sup:38r0fai0]

so

300=N[sub:38r0fai0]0[/sub:38r0fai0]e[sup:38r0fai0]20k[/sup:38r0fai0]

but by what method did you come up with

N[sub:38r0fai0]0[/sub:38r0fai0]=300e[sup:38r0fai0]-20k[/sup:38r0fai0]

I dont understand how the N[sub:38r0fai0]0[/sub:38r0fai0] and the term 300 were able to trade places, and why the exponent was turned negative. And also what rule allows you to substitute that into this equation

1200=N[sub:38r0fai0]0[/sub:38r0fai0]e[sup:38r0fai0]40k[/sup:38r0fai0]

to come up with

1200=300e[sup:38r0fai0]20k[/sup:38r0fai0]

 

[Mrspi]

thank you so much

i have a couple clarifying questions though.
N=N[sub:3aabehg7]0[/sub:3aabehg7]e[sup:3aabehg7]tk[/sup:3aabehg7]

so

300=N[sub:3aabehg7]0[/sub:3aabehg7]e[sup:3aabehg7]20k[/sup:3aabehg7]

but by what method did you come up with

N[sub:3aabehg7]0[/sub:3aabehg7]=300e[sup:3aabehg7]-20k[/sup:3aabehg7]

I dont understand how the N[sub:3aabehg7]0[/sub:3aabehg7] and the term 300 were able to trade places, and why the exponent was turned negative. And also what rule allows you to substitute that into this equation

1200=N[sub:3aabehg7]0[/sub:3aabehg7]e[sup:3aabehg7]40k[/sup:3aabehg7]

to come up with

1200=300e[sup:3aabehg7]20k[/sup:3aabehg7]

Ok...let's take it step by step.

300=N[sub:3aabehg7]0[/sub:3aabehg7]* e[sup:3aabehg7]20k[/sup:3aabehg7]

Divide both sides of the equation by e[sup:3aabehg7]20k[/sup:3aabehg7]

300 / e[sup:3aabehg7]20k[/sup:3aabehg7] = N[sub:3aabehg7]0[/sub:3aabehg7]

Now remember the definition of a negative exponent: 1 / a[sup:3aabehg7]n[/sup:3aabehg7] = a[sup:3aabehg7]-n[/sup:3aabehg7]

So, 300 / e[sup:3aabehg7]20k[/sup:3aabehg7] = 300 * e[sup:3aabehg7]-20k[/sup:3aabehg7]

Ok?

 

[brianwolk]

ok, thank you guys.