Question

A new product is to be test marketed by giving it free to 1000 people in a city of one million inhabitants, which is assumed to remain constant for the period of the test. It is further assumed that the rate of product adoption will be proportional to the number of people who have it with the number who do not. The proportionality constant is 0.275×10⁻⁶=0.275E(−6). Estimate as a function of time the number of people who will adopt the product if it known that 3000 people have adopted the product after four weeks.

Answer 1

The estimated number of people who will adopt the product as a function of time,
`\(N(t)\)`, after four weeks can be expressed as
`\(N(t) = (3000)/(1 + 0.275 * 10^(-6) * (1,000,000 - 3000) * t)\)`, where
`\(t\)` is the time in weeks.

Step-by-step explanation:

In the given scenario, the rate of product adoption is assumed to be proportional to the difference between the number of people who have the product and those who do not. The proportionality constant is
`\(0.275 * 10^(-6)\)`. The formula
`\(N(t)\)` represents the estimated number of people adopting the product at time
`\(t\)`.

The key assumption is that the rate of adoption is proportional to the difference between the population and the number of adopters, leading to a differential equation of the form
`\((dN)/(dt) = k * (1,000,000 - N)\)`, where
`\(k\)` is the proportionality constant. Solving this differential equation yields the expression for
`\(N(t)\)`.

After four weeks
`(\(t = 4\))`, the given information is
`\(N(4) = 3000\)`. Substituting this into the formula and solving for the proportionality constant
`\(k\)`, we arrive at the final expression. This formula provides an estimate of the number of people adopting the product over time, taking into account the assumed proportional adoption rate. The relationship is inversely proportional to the remaining non-adopting population, indicating a slowing adoption rate as more people acquire the product.