Problem 92 - Intermediate Algebra

Question

Wikipedia. The number of articles \(a(t)\) in millions in the Englishlanguage edition of Wikipedia can be approximated by the function \(a(t)=0.54 t+0.5,\) where \(t\) is the number of years since \(2005 .\) Use an inequality to determine those years for which the number of articles surpassed 2.5 million. (Source: Wikipedia article: Size of Wikipedia)

Step-by-Step Solution

Answer

The articles surpassed 2.5 million in the year 2009.

Step 1: Identify the given function

The given function that approximates the number of articles on Wikipedia is \( a(t) = 0.54t + 0.5 \). Here, \( a(t) \) represents the number of articles in millions, and \( t \) is the number of years since 2005.

Step 2: Set up the inequality

We need to determine the values of \( t \) for which the number of articles surpassed 2.5 million. Therefore, we set up the inequality: \[ 0.54t + 0.5 > 2.5 \]

Step 3: Solve the inequality

Subtract 0.5 from both sides of the inequality:\[ 0.54t > 2.0 \]Next, divide both sides by 0.54 to solve for \( t \): \[ t > \frac{2.0}{0.54} \] This simplifies to approximately \( t > 3.70 \).

Step 4: Interpret the solution

Since \( t \) represents the number of years since the year 2005, add 3.70 to 2005 to find the year: 2005 + 3.70 gives us approximately 2008.70. Therefore, the articles surpassed 2.5 million sometime during the year 2009.

Key Concepts

Function Approximation

Function approximation is an important concept when dealing with real-world data. Often, complex data sets can be simplified using mathematical functions that closely mimic the data trends. In our context, we use a linear function to approximate the growth in the number of Wikipedia articles over time.

In the exercise given, the function is defined as:

• \(a(t) = 0.54t + 0.5\)
• Here, \(a(t)\) estimates the millions of articles in Wikipedia, and \(t\) represents the number of years passed since 2005.

Linear functions are chosen for their simplicity and ease of use, particularly when initial data shows consistent increase or decrease. The equation \(0.54t + 0.5\) suggests a steady annual growth rate of 0.54 million articles. Function approximation helps transform abstract data into understandable formats making it accessible for practical applications, like making predictions in this exercise.

Solving Inequalities

Solving inequalities involves finding all possible values of a variable that satisfy a given condition. In the example, the task was to determine when the number of Wikipedia articles surpassed 2.5 million. The inequality established was:

• \(0.54t + 0.5 > 2.5\)

Here are the steps to solve it:

• First, subtract 0.5 from both sides to isolate terms with \(t\) on one side: \[0.54t > 2.0\]
• Next, divide each side by 0.54 to solve for \(t\):\[t > \frac{2.0}{0.54}\]
• After simplifying, the solution is approximately \(t > 3.70\).

These steps demonstrate how inequalities can model situations where you wish to find values that exceed a certain threshold. Remember, solving inequalities requires maintaining the inequality's direction unless multiplying or dividing by negatives, which isn't the case here.

Graphical Interpretation

Graphs offer a visual representation of mathematical functions and inequalities, aiding in intuitive understanding. In our scenario, the linear function \(a(t) = 0.54t + 0.5\) can be graphed as a straight line. This line allows us to visually infer when the number of articles exceeds 2.5 million.

Here's how you interpret graphically:

• Start by plotting \(t\) on the x-axis (years since 2005) and \(a(t)\) on the y-axis (articles in millions).
• The line intersects the y-axis at 0.5 and has a positive slope of 0.54, indicating consistent growth.
• Draw a horizontal line at \(a(t) = 2.5\) million. The intersection point of this line with the function \(a(t) = 0.54t + 0.5\) provides a tangible estimate when \(t > 3.70\).

Graphical interpretation makes it easier to understand how changes in one variable impact another, reinforcing insights gained through algebraic calculations. This way, both visual and numerical clarity is achieved, enhancing comprehension and practical application of inequalities.