Problem 26 - Applied Calculus

Question

The number of US households with cable television \(^{40}\) was 12,168,450 in 1977 and 73,365,880 in \(2003 .\) Estimate the average rate of change in the number of US households with cable television during this 26 -year period. Give units and interpret your answer.

Step-by-Step Solution

Answer

The average rate of change was approximately 2,353,747 households per year.

Step 1: Understanding the Problem

We need to calculate the average rate of change in the number of US households with cable television from 1977 to 2003. This equates to finding how much the number changed per year over this period of 26 years.

Step 2: Identify the Data Points

The number of households in 1977 was 12,168,450, and in 2003 it was 73,365,880. The time period over which this change occurred is from 1977 to 2003, a total of 26 years.

Step 3: Calculate the Change in Number of Households

Subtract the number of households in 1977 from the number in 2003: \[ 73,365,880 - 12,168,450 = 61,197,430 \] This represents the total change in the number of households with cable television.

Step 4: Calculate the Average Rate of Change

Divide the total change by the number of years (26) to find the average rate of change: \[ \frac{61,197,430}{26} = 2,353,747.31 \] This means the average change in the number of households with cable television per year.

Step 5: Interpret the Answer

The average rate of change in the number of US households with cable television from 1977 to 2003 was approximately 2,353,747 households per year. This reflects an increase in the number of households with cable television.

Key Concepts

US Households

In the context of this problem, let's focus on the role of US households. The term "household" refers to a group of people, often a family, who live together in the same residence, sharing living spaces and resources. When examining how many households have cable television, this information helps us understand the popularity and reach of cable services among the US population.

In 1977, there were approximately 12 million US households with cable TV, which increased to over 73 million by 2003. Tracking changes in household technology adoption can reveal societal trends and economic growth. This increase during the time period might suggest more widespread access to technology, changing consumer preferences, or the affordability of cable services.

Understanding the behavior of US households helps us to see how amenities like cable TV become common, shaping communication and entertainment landscapes.

Cable Television

Cable television in the 1970s to early 2000s experienced significant growth, reflected in the sharp increase in US households having cable TV. Cable television systems distribute television programming to consumers via coaxial or fiber-optic cables, as opposed to traditional broadcasting over airwaves. This method can offer higher quality and more channels.

The surge in cable TV adoption can be attributed to several factors:

• Expanded channel offerings including specialized programming.
• Improved picture quality over traditional broadcast means.
• Access to premium content, including sporting events and newly-released films.

As more Americans began to embrace cable TV, it paved the way for the modern telecommunications landscape, including the development of broadband internet and digital cable services. It provides an essential backdrop for understanding how technology and entertainment preferences shifted over decades.

Calculus

Calculus is at the heart of understanding changes like those seen in the number of households adopting cable TV. Specifically, we use the concept of the average rate of change, which is a fundamental idea in calculus. In simpler terms, it measures how something changes over a certain period of time.

To find this, we take two key data points: the starting point (12,168,450 households in 1977) and the ending point (73,365,880 households in 2003). By assessing the total change (61,197,430 households) over the 26-year period, we calculate how many households, on average, started using cable TV per year. This calculation is done by dividing the total change by the number of years: \[ \text{Average Rate of Change} = \frac{{73,365,880 - 12,168,450}}{{26}} = 2,353,747.31txt{households per year}\]

Using calculus to assess average rate of change provides valuable insights into trends and allows for informed predictions about future behavior.