# Math Help - Norwegian Integral Highschool task

1. ## Norwegian Integral Highschool task

" A germpopulation have on a given time 10^5 germ. N = germnumber after h, hours and assume that the growingspeed is given with N= 0,15N(1-10^-6N)."

a) Show that the diff.equation is separable.

b) Find N(h) and draw the graph.

c) Find the germnumber after 5 hour.

d) When do the germnuber grow fastest?

e) What are the name of this type of modell? (function/growingmoddel)

2. dN/dh = 0,15N(1-10^-6N)

It is fairly simple to separate

dN/(0,15N(1-10^-6N)) = dh

Integrate by parts.

Once you Have N(h) c is easy

This is an example of a logistic differential equation

" A germpopulation have on a given time 10^5 germ. N = germnumber after h, hours and assume that the growingspeed is given with N= 0,15N(1-10^-6N)."

a) Show that the diff.equation is separable.

b) Find N(h) and draw the graph.

c) Find the germnumber after 5 hour.

d) When do the germnuber grow fastest?

e) What are the name of this type of modell? (function/growingmoddel)
$k = 1.5 \times 10^{-7}$

$M = 10^6$

$\frac{dN}{dh} = kN(M - N)$

$\frac{dN}{N(M-N)} = k \, dh$

$\frac{1}{M} \left(\frac{1}{N} + \frac{1}{M-N}\right) \, dN = k \, dh$

$\int \left(\frac{1}{N} - \frac{-1}{M-N}\right) \, dN = Mk \int \, dh$

$\ln{N} - \ln(M-N) = MKh + C$

$\ln\left(\frac{N}{M-N}\right) = Mkh + C$

$\frac{N}{M-N} = Ae^{Mkh}$

$N = \frac{M}{1 + Be^{-Mkh}}$

$10^5 = \frac{10^6}{1 + B}$

$N = \frac{M}{1 + 9e^{-Mkh}}$

4. Thanks !

but can you please show me d) ?

Thanks !

but can you please show me d) ?
Find the t-coordinate of the point of inflection of N.

6. N= 0,15N(1-10^-6N)

N= 0,15N-1,5*10^-7N^2

N= 0,15-3*10^-7N

N= 0

0,15 = 3*10^-7N

N = 500000

Grow fastes when N(t) = 500000

Is this right?