Growth problem

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.

Last edited by a moderator:

cristo
Staff Emeritus
Your first equation is not visible. Perhaps you could rewrite it?

ranger
Gold Member
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:

$$\frac{dP}{dt} = \frac {6}{700}P(7-P)$$

cristo
Staff Emeritus
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:

$$\frac{dP}{dt} = \frac {6}{700}P(7-P)$$
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: $$\frac{dP}{dt} = \frac {6}{700}P(7-P)$$. 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: $$\int \frac{dP}{P(7-P)}=\int\frac{6}{700}dt +C$$

Do you know how to solve this?

Last edited by a moderator:
ranger
Gold Member
You did put +C accidentally right?

Last edited:
Yeah, I can solve that. Didn't think to seperate variables for some retarded reason. Thanks for the help.

cristo
Staff Emeritus
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!

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.

cristo
Staff Emeritus
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