91Ó°ÊÓ

Use a calculator with an \(\overline{e^{x}}\). key to solve. As of July \(2010,500\) million people worldwide shared versions of their lives on Facebook. The graph shows the number of active Facebook users (users who returned to the site within 30 days) for selected months from 2009 through 2010. (BAR GRAPH CAN'T COPY) The data can be modeled by $$f(x)=19 x+127 \quad \text { and } \quad g(x)=152.6 e^{0.0667 x}$$ in which \(f(x)\) and \(g(x)\) represent the number of active Facebook users, in millions, \(x\) months after December 2008 . a. According to the linear model, how many millions of active Facebook users were there in July 2010 , 19 months after December \(2008 ?\) b. According to the exponential model, how many millions of active Facebook users were there in July \(2010 ?\) c. Which function is a better model for the data in July \(2010 ?\)

Step by step solution

01

Substitute in the linear model

We first substitute \(x = 19\) in \(f(x) = 19x + 127\). Doing this calculation, we find that \(f(19) = 361\) million Facebook users.

02

Substitute in the exponential model

Next, we substitute \(x = 19\) in \(g(x) = 152.6e^{0.0667x}\). Using a calculator with the \(e^{x}\) function to compute this, we find \(g(19)\) is approximately \(500\) million Facebook users.

03

Comparison

We compare the two results. The linear model gives 361 million users while the exponential model gives 500 million users. We know that in July 2010 there were 500 million Facebook users, therefore the exponential model is a better fit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Models

Linear models are mathematical tools used to describe a straight-line relationship between variables. In the context of the problem, a linear model is used to estimate the number of active Facebook users over time. The linear function given in the exercise is \(f(x) = 19x + 127\), where \(x\) represents the number of months after December 2008. Here, the slope of the line is 19, indicating that on average, Facebook user numbers increase by 19 million each month.

• The advantage of a linear model is its simplicity and ease of use. It is great for making quick calculations.
• However, linear models might not always provide the most accurate predictions, especially over longer periods when growth rates might change.

In this problem, the linear model predicted 361 million users for July 2010. This number is calculated simply by substituting \(x = 19\), a straightforward and quick method.

Facebook User Statistics

Facebook user statistics help us understand how social media demographics change over time. In 2010, Facebook had around 500 million active users, which is a huge number and indicative of a fast-growing platform. This rapid growth is what makes Facebook statistics particularly interesting.

• Active users are defined as those who return to the site within 30 days, which means they are regularly engaged with the platform.
• Statistics like these are crucial for understanding audience behavior and can be used to predict future trends.

Comparing models can show how different assumptions about growth can lead to different projections. In the given problem, the actual count of 500 million users aligns with the exponential model, demonstrating the platform's rapid growth and how quickly digital culture can change.

Mathematical Modeling

Mathematical modeling involves creating abstract models that explain complex systems using mathematical language. In this exercise, both linear and exponential models are used to represent Facebook's growth in users.

• The choice of model can impact the accuracy of predictions. Exponential models are often used when growth rates are not constant but instead grow over time, such as in population growth or the spread of technologies like Facebook.
• The exponential model used is \(g(x) = 152.6e^{0.0667x}\), where \(e\) is the base of natural logarithms, suggesting a more complex, non-linear growth pattern.

Mathematical modeling helps us understand various "what-if" scenarios and can be an essential tool for decision-making and planning. In the given context, it allowed understanding of the growth pattern of Facebook and choosing the exponential model as more suitable based on actual user data.