Question

The number N(t) of supermarkets throughout the country that are using a computerized checkout system is described by the initial-value problem

dN/dt = N(1 − 0.0008N), N(0) = 1.

Required:
Predict how many supermarkets are expected to adopt the new procedure over a long period of time?

Answer 1

Answer:

1250 supermarkets

Step-by-step explanation:

Given

$\frac{dN}{dt} = N(1 - 0.0008N)$

$N(0) = 1$

Required

Number of supermarkets over a long period of time

To do this, we simply set

$\frac{dN}{dt} = 0$

So, we have:

$N(1 - 0.0008N) = 0$

Split

$N = 0\ or\ 1 - 0.0008N = 0$

Solve: $1 - 0.0008N = 0$

Collect like terms

$0.0008N = 1$

Make N the subject

$N = \frac{1}{0.0008}$

$N = 1250$

So:

$\lim_{t \to \infty} N(t) = 1250$