Problem 10 - Applied Calculus

Question

From October 2002 to October 2006 the number \(N(t)\) of Wikipedia articles was approximated by \(N(t)=\) \(N_{0} e^{t / 500}\), where \(t\) is the number of days after October 1,2002 . Find the doubling time for the number of Wikipedia articles during this period.

Step-by-Step Solution

Answer

The doubling time is approximately 346.5 days.

Step 1: Understand the Problem

We need to find the doubling time for the number of Wikipedia articles. This means we need to determine when the number of articles becomes twice its original amount, \(N_0\).

Step 2: Set Up the Equation for Doubling Time

The initial number of articles is \(N_0\). When the articles double, we have \(2N_0 = N_0 e^{t/500}\). Our goal is to solve this equation for \(t\).

Step 3: Divide Both Sides by \(N_0\)

To simplify the equation, divide both sides by \(N_0\): \(2 = e^{t/500}\). This isolates the exponential term.

Step 4: Take Natural Logarithm on Both Sides

Apply the natural logarithm to both sides of the equation to solve for \(t\): \(\ln(2) = \frac{t}{500}\).

Step 5: Solve for \(t\)

Multiply both sides of the equation by 500 to find \(t\): \(t = 500 \ln(2)\).

Step 6: Calculate the Doubling Time

Use the approximate value \(\ln(2) \approx 0.693\) to calculate \(t\): \(t = 500 \times 0.693 = 346.5\). Thus, the doubling time is approximately 346.5 days.

Key Concepts

Doubling Time

Doubling time is the period it takes for a quantity to double in size or value. It's a fundamental concept in exponential growth, which is often observed in populations, investments, and digital data growth like Wikipedia articles.

• When you have a quantity that grows exponentially, like the number of Wikipedia articles given by the equation \(N(t) = N_0 e^{t/500}\), it doubles over consistent intervals.
• To find the doubling time, you need the time \(t\) for the number of articles to hit \(2N_0\). Setting \(2N_0 = N_0 e^{t/500}\) allows finding this interval.
• After simplifying, this gives \(2 = e^{t/500}\), a simpler equation to handle for calculating \(t\).

Understanding doubling time is essential to predict how quickly the quantity spirals into large numbers, which is crucial for long-term planning and comprehension.

Natural Logarithm

The natural logarithm, represented as \(\ln\), is logarithm to the base \(e\), where \(e\) is approximately 2.71828. It's a vital tool in calculus and exponential equations.

• In the equation \(2 = e^{t/500}\), taking the natural log of both sides gives \(\ln(2) = \frac{t}{500}\).
• This operation simplifies exponential equations, transforming multiplication into addition, which is easier to solve.
• Natural logs are extensively used to manipulate expressions involving growth, decay, and compounding processes.

When solving problems like finding doubling time, the natural log lets you extract the exponent, giving precise and computable results.

Calculus Problem Solving

Problem-solving in calculus often involves understanding and manipulating mathematical models to find solutions. Let's uncover how this approach applied to our exercise:

• Identify the problem and set up an equation: We began with the exponential function describing Wikipedia's article growth.
• Isolate terms: We isolated the exponent by dividing both sides by \(N_0\), which reduced complexity.
• Utilize natural logarithms: This critical step simplified our latest equation to find \(t\) with minimal computation.
• Finally, solve for the variable: Calculating \(t = 500 \ln(2)\) provided the estimated time for the articles to double.

Approaching calculus problems systematically not only enhances solution accuracy but also builds a deeper understanding of mathematical relationships.