Problem 15 - Applied Calculus

Question

An early model of the growth of the Wikipedia assumed that every day a constant number, \(B\), of articles are added by dedicated Wikipedians and that other articles are created by the general public at a rate proportional to the number of articles already there. Express this model as a differential equation for \(N(t),\) the total number of Wikipedia articles \(t\) days after it started on January 15 2001

Step-by-Step Solution

Answer

The differential equation is \(\frac{dN}{dt} = B + kN(t)\).

Step 1: Understanding the Problem

We need to model the growth of Wikipedia articles using two sources: a constant addition of articles by dedicated Wikipedians, and a rate proportional to the existing number of articles from the general public. The solution is to derive a differential equation for the total number of articles over time, \(N(t)\).

Step 2: Identify Constant Growth Part

The constant addition part is straightforward: if \(B\) articles are added by dedicated Wikipedians each day, this contributes a constant term \(B\) in our differential equation.

Step 3: Formulate Proportional Growth Part

For the part added by the general public, which is proportional to the existing number of articles, we introduce a constant of proportionality \(k\). Therefore, the number of articles added by the general public per day can be expressed as \(kN(t)\).

Step 4: Combine Both Growth Parts

The total rate of change in the number of articles, \(\frac{dN}{dt}\), consists of both the constant addition and the proportional addition. This gives the differential equation:\[ \frac{dN}{dt} = B + kN(t) \]

Step 5: Conclusion of the Model

The differential equation \(\frac{dN}{dt} = B + kN(t)\) describes how the total number of Wikipedia articles grows over time, taking into account both the constant rate of dedicated contributions and the proportional growth from the general public.

Key Concepts

Exponential Growth Model

The concept of the exponential growth model is important for understanding dynamic processes that grow rapidly over time. It is characterized by a growth rate that becomes increasingly rapid in relation to the growing total number or size. In the context of Wikipedia's growth, this model can help describe how the platform expands due to the number of articles being added. When we say that something is growing exponentially, it means that the rate of growth is directly proportional to the current amount. For instance, if the number of articles on Wikipedia increases, the speed of its growth (in terms of new article generation) gets faster, as per the general public's contributions described in the exercise. This kind of growth can often be described by differential equations, where the rate of change is not linear but instead influenced by the total quantity at any given time. As the number of articles becomes larger, the contribution from proportional growth through public participation becomes more significant. This results in the rapid acceleration of article numbers over time.

Constant Rate

A constant rate of growth means that the number of items added does not change over time. In our Wikipedia growth model, this is represented by the dedicated Wikipedians who add a fixed number of articles every day. This constant addition is a straightforward concept: each day, no matter what else happens, the same number of articles are added. In mathematical terms, this is reflected in our differential equation as a constant term, denoted by **B**. The impact of a constant rate can seem small when examined in isolation. However, when combined with other growth factors, such as proportional growth, it contributes to the overall increase in the total number over time. This makes the differential equation truly representative of real-world scenarios where multiple factors influence growth simultaneously.

Proportional Growth

Proportional growth occurs when the growth rate depends directly on the current amount present. In the exercise, this means that additional articles are created by the general public at a rate proportional to how many articles already exist. This is expressed mathematically by introducing a constant of proportionality, represented by **k** in the differential equation. The greater the number of articles, the larger the addition made by this type of growth. Thus, as Wikipedia's base number of articles increases, more articles are added through public contributions. This approach is vital as it highlights how community-driven efforts can scale with the success of a platform. The larger Wikipedia gets, the more energy and resources it attracts from users worldwide. Models incorporating proportional growth help us predict such phenomena accurately, reflecting realities where existing size or number enhances further growth.