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

Long-term prediction for the number of supermarkets adopting a computerized checkout system, based on the logistic growth model provided, leads to a carrying capacity of 1250 supermarkets as time approaches infinity.

Step-by-step explanation:

The differential equation given describes logistic growth, which is a common way to model population dynamics where growth is limited by certain factors, such as resource limitations. In this case, the population dynamics of supermarkets adopting a computerized checkout system is being modeled.

To find the long-term prediction for the number of supermarkets adopting the system (N(t)), we need to find the carrying capacity, which is when the growth rate dN/dt equals zero, leading to no further increase in N. Setting the growth equation equal to zero and solving for N yields:

dN/dt = N(1 - 0.0008N) = 0

1 - 0.0008N = 0

0.0008N = 1

N = 1/0.0008

N = 1250

As t approaches infinity, the number of supermarkets N(t) is expected to approach the carrying capacity, which is 1250 supermarkets.

Answer 2

1250 supermarkets

Step-by-step explanation:

Given

`(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

`(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 = (1)/(0.0008)`

`N = 1250`

So:

`\lim_(t \to \infty) N(t) = 1250`