Number 5 The number N(t) of supermarkets throughout a country that are using a c

Question

Number 5 The number N(t) of supermarkets throughout a country that are using a computerized checkout system is described by the initial value logistic equation 4. dN N(1 0.0005N), How many supermarkets are expected to adopt the new technology when t 10? 5. The number N(t) of people in a community who are exposed to a certain advertisement after t days is described by the logistic equation dt in which, M is the Saturation level (Carrying capacity) and k is a proportionality constant to be determined. The advertisement is initially exposed to 10 people in the community and it is predicted that the limiting number of people who will be exposed to this ad within the community is 100. If it is observed that N(1) 25, find the time taken for 50 people to be exposed to this advertisement. (Hint: First find the proportionality constant k)

Explanation / Answer

dN/dt=k*(M-N)*N

==>dN/(N*(M-N))=k*dt

==>(1/N)*dN+(1/(M-N))*dN=(k*M)*dt

==>ln(N)-ln(M-N)=k*M*t

==> ln(N/(M-N))=k*M*t+c

given M=100

N(0)=10

N(1)=25

at t=0, N=10

c=ln(10/90)=-2.1972

at t=1,N=25

then ln(25/(100-25))=k*100*1-2.1972

==>k=0.010986

so let at time t=T, N=50

then ln(N/(100-N))=0.010986*T-2.1972

==>T=200 days

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