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 January 1, 2001 .

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We are tasked with creating a model for the growth of Wikipedia articles over time. The problem specifies two sources of article growth: a constant addition of articles and a growth rate proportional to the current number of articles.

Step 2: Define the Known Quantities and Variables

Let \(N(t)\) denote the total number of articles after \(t\) days. The problem states that a constant number \(B\) of articles is added daily by wikipedians, and articles are added at a rate proportional to \(N(t)\) by the public.

Step 3: Express the Growth Equation

The total rate of change of the number of articles, \( \frac{dN}{dt} \), will be the sum of the constant rate \(B\) and the rate proportional to \(N(t)\), which can be expressed as \(kN(t)\), with \(k\) being the proportionality constant.

Step 4: Formulate the Differential Equation

Combine the constant rate of change and the proportional rate of change into one expression: \[ \frac{dN}{dt} = B + kN(t) \]. This differential equation represents the given model of Wikipedia's growth.

Step 5: Interpret the Equation

The equation \( \frac{dN}{dt} = B + kN(t) \) captures both the constant addition of articles and the growth proportional to the current number. This formulation sets the framework for solving for \(N(t)\) given initial conditions and parameters \(B\) and \(k\).

Key Concepts

Growth Models

When we talk about growth models in the context of Wikipedia's article development, we're essentially discussing how a system expands or evolves with time. A growth model offers a mathematical representation of this progression.
Consider the Wikipedia example: initially, there are zero articles on January 1, 2001. Over time, dedicated contributors and the general public add articles. This is where the growth model comes in to explore and predict this pattern. A crucial aspect of growth models is that they allow us to dissect the different components that contribute to growth.
This may include:

• A constant increase, such as a fixed number of articles added by dedicated editors each day. This is stable and predictable, providing a steady stream of growth.
• A variable increase, like articles added by the general public at a rate that depends on how many articles are already present. This introduces a dynamic component to the growth model.

By understanding the growth trends and factors at play, a growth model helps forecast future changes and guide decisions on content development.

Rate of Change

The concept of the "rate of change" tells us how fast a quantity is increasing or decreasing.
In Wikipedia's growth context, this is represented by the rate at which new articles are added. The rate of change is mathematically expressed as a derivative, noted as \( \frac{dN}{dt} \), which describes how the number of articles evolves over time.There are two main components to Wikipedia's rate of change:

• The constant rate \( B \), representing the daily addition of articles by dedicated Wikipedians. It's a fixed contribution, consistently adding to the total number of articles.
• The proportional rate \( kN(t) \), which varies depending on the existing number of articles. Here, \( k \) is the proportionality constant, showing how the article count influences further growth.

Combining these creates the overall rate of change and gives us the differential equation:\[ \frac{dN}{dt} = B + kN(t) \].
This equation helps us visualize the balance between constant additions and the influence of the current article count on future growth.

Proportional Relationships

Proportional relationships are key in understanding how certain growth aspects work. In simple terms, a proportional relationship implies that one quantity changes at a rate that is either directly or inversely related to another.In the Wikipedia growth model, the contribution from the public is not fixed like the daily additions by Wikipedians. Instead, it's proportional to the existing number of articles \( N(t) \).
This means the more articles there are, the more frequently new ones will be added by the public, making growth more dynamic.The proportional rate is represented by \( kN(t) \):

• \( k \) is the constant of proportionality, determining how sensitively the number of new articles responds to the current article count.
• As \( N(t) \) increases, so does the rate of article creation by the public, implying a snowball effect where growth accelerates over time.

This concept is crucial because it captures how the existing size of Wikipedia impacts its future development, highlighting the compound nature of growth driven by proportional relationships.