Problem 56 - Algebra 1: Concepts and Skills

Question

Use the following information. An Internet service provider estimates that the number of households \(h\) (in millions) with Internet access can be modeled by the equation \(h=6.76 t+14.9\) where \(t\) represents the number of years since 1996. What does the graph mean in the context of the real-life situation?

Step-by-Step Solution

Answer

The graph of the given equation shows the predicted growth in the number of households with Internet access since 1996. The starting point is 14.9 million households in 1996, with an increase rate of 6.76 million households per year.

Step 1: Identify the Variables

The given equation is \(h=6.76t+14.9\). In this equation, \(h\) represents the number of households with Internet access in millions, while \(t\) represents the number of years elapsed since 1996.

Step 2: Determine the Meaning of Coefficients

In the equation, 6.76 is the coefficient of \(t\), meaning that for each passing year since 1996, the estimated number of households with Internet access increases by 6.76 million. On the other hand, 14.9 is the y-intercept, it represents the estimated number of households with internet access in 1996, the year when \(t=0\).

Step 3: Explain the Graph in Real-Life Context

The line graph of this equation in the context of the real-life situation signifies how the number of households with Internet access has increased over the years. Starting from an estimated 14.9 million households in 1996 and increasing at a rate of 6.76 million households per year.

Key Concepts

Internet Usage

In the modern world, internet usage has become a fundamental part of our daily lives and socio-economic development. The concept of internet usage often refers to how people and households access and utilize the internet for various purposes like communication, education, and entertainment.
Understanding internet usage can provide insights into technological growth and access equity around the world.

• As more households gain access to the internet, global connectivity improves.
• Internet usage influences economic opportunities, education access, and information distribution.

This exercise specifically refers to the rise in the number of households gaining internet access, representing a trend that underscores the broader implications of increased connectivity. By observing how households adopt the internet, we can better understand societal growth patterns, digital divide issues, and resource allocation.

Household Internet Access

Household internet access indicates how many homes are connected to the internet. The ability to access the internet from home is crucial in bridging the digital divide, which is the gap between those who have ready access to computers and the internet, and those who do not.
Access to the internet enhances everyday living and learning.

• It enables easier communication between family and peers.
• It provides access to a wealth of information from educational resources to news and streaming services.

In the problem outlined, a mathematical model helps estimate how household internet access has grown over the years. This allows stakeholders to understand past trends and predict future demands for internet services. As we aim to make internet access universal, tracking these statistics is vital for policy-makers and service providers to mitigate access barriers.

Slope Interpretation

In mathematical modeling, the slope signifies the rate at which a dependent variable changes relative to an independent variable. In linear equations like the one discussed in the exercise, the slope is the coefficient of the variable.
Here, the equation is given by \( h = 6.76t + 14.9 \). The slope in this context is 6.76, which can be interpreted as:

• The rate at which the number of households with internet access increases each year.
• For each year that passes after 1996, 6.76 million new households gain internet access.

Understanding this concept is crucial as it presents the growth trend. By interpreting slopes, one can visualize how quickly or slowly changes occur over time. Such interpretations help in making informed projections and decisions, especially related to infrastructure and resource allocation in rapidly evolving technological environments.