Growth problem

  • Thread starter glid02
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  • #1
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Here is the first question:
A population P obeys the logistic model. It satisfies the equation
https://webwork.math.uga.edu/webwork2_files/tmp/equations/11/02d0a645c053f1f6002d746c78143f1.png

Assume that P(0)=3. Find P(66)

First I multiplied both sides by dt and integrated, giving:
P=6/700Pt(7-P)+c
If P(0)=3 then c=3
P=6/700Pt(7-P)+3

Then I divided everything by P and had
1=6/700t(7/P-1)+3/P

Now to find P(66)
1=6/700*66(7/P-1)+3/P
1=396/700(7/P-1)+3/P
1=2772/700P-396/700+3/P
4872/700P=1096/700
P=4.445

That's not right, what am I missing?
Thanks.
 
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Answers and Replies

  • #2
cristo
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Your first equation is not visible. Perhaps you could rewrite it?
 
  • #3
ranger
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You have to accept some sort of web certificate to view the equation. It seems to be located on some university's website. Heres the equation that I get:

[tex]\frac{dP}{dt} = \frac {6}{700}P(7-P)[/tex]
 
  • #4
cristo
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You have to accept some sort of web certificate to view the equation. It seems to be located on some university's website. Heres the equation that I get:

[tex]\frac{dP}{dt} = \frac {6}{700}P(7-P)[/tex]
Ahh, ok, thanks for that, ranger. I must have clicked no automatically!

Here is the first question:
A population P obeys the logistic model. It satisfies the equation
https://webwork.math.uga.edu/webwork2_files/tmp/equations/11/02d0a645c053f1f6002d746c78143f1.png

Assume that P(0)=3. Find P(66)

First I multiplied both sides by dt and integrated, giving:
P=6/700Pt(7-P)+c
If P(0)=3 then c=3
P=6/700Pt(7-P)+3

Then I divided everything by P and had
1=6/700t(7/P-1)+3/P

Now to find P(66)
1=6/700*66(7/P-1)+3/P
1=396/700(7/P-1)+3/P
1=2772/700P-396/700+3/P
4872/700P=1096/700
P=4.445

That's not right, what am I missing?
Thanks.
You have this equation: [tex]\frac{dP}{dt} = \frac {6}{700}P(7-P)[/tex]. You cannot simply multiply by dt and integrate, since you have not integrated the terms including P wrt P! You must rearrange the equation to give: [tex]\int \frac{dP}{P(7-P)}=\int\frac{6}{700}dt +C[/tex]

Do you know how to solve this?
 
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  • #5
ranger
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You did put +C accidentally right?
 
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  • #6
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Yeah, I can solve that. Didn't think to seperate variables for some retarded reason. Thanks for the help.
 
  • #7
cristo
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Science Advisor
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You did put +C accidentally right?
Yea, I guess I haven't really integrated anything yet, so strictly the constant doesn't appear until the next line!

Yeah, I can solve that. Didn't think to seperate variables for some retarded reason. Thanks for the help.
You're welcome!
 
  • #8
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OK I lied, I'm still not getting the right answer.

dP/P(7-P)=6/700dt
1/7log(P)-1/7log(7-P)=6t/700+c
log(P)-log(7-P)=6t/100+c
Using e
P-7+P=e^(6t/100)+c
P=(e^(6t/100)+7)/2+c
P(0)=3 so c=-1
Subbing 66 for t, I get 28.729

Still not right, what am I doing wrong now?
Thanks again.
 
  • #9
cristo
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OK I lied, I'm still not getting the right answer.

dP/P(7-P)=6/700dt
1/7log(P)-1/7log(7-P)=6t/700+c
log(P)- log(7-P)=6t/100+c
The - sign in red should be a +
Using e
P-7+P=e^(6t/100)+c
What you've done here is wrong. You must collect the logarithmic terms before you can take the exponential of both sides.
 
  • #10
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You lost me with the collecting. Can you give me another example?
 
  • #11
cristo
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Science Advisor
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Well, have you come across the general rule: log(a)+log(b)=log(ab) ?
 
  • #12
54
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If I had I'd forgotten it. I should be able to solve from here (again). Thanks again for helping.
 

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