Problem 30 - Applied Calculus

Question

The number of US households with cable television \(^{39}\) 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 is approximately 2,353,747 households per year.

Step 1: Identify Known Values

First, identify the values given in the problem. We know the number of households was \(12,168,450\) in \(1977\) and \(73,365,880\) in \(2003\). The duration between these two years is \(26\) years.

Step 2: Understand the Concept of Average Rate of Change

The average rate of change can be found by taking the difference in the value of the quantities over the time period. It is essentially the slope of the line connecting two points in a time series.

Step 3: Calculate the Change in Number of Households

Subtract the number of households in 1977 from the number in 2003: \(73,365,880 - 12,168,450 = 61,197,430\).

Step 4: Calculate the Average Rate of Change

Now, divide the change in the number of households by the number of years:\[\text{Average Rate of Change} = \frac{61,197,430}{26} = 2,353,747 \text{ households per year.}\]

Step 5: Interpret the Result

The average rate of change of \(2,353,747\) households per year means that, on average, the number of US households with cable television increased by approximately \(2,353,747\) each year from 1977 to 2003.

Key Concepts

Cable Television Growth

The growth of cable television in the United States over the period from 1977 to 2003 is a remarkable example of technological adoption. During these 26 years, the number of US households with cable television increased dramatically. Such growth illustrates a significant shift in consumer preferences and the increasing penetration of cable services across the country. Key points to understand about cable television growth include:

• The adoption of cable television brought a diverse array of channels and content directly to households, which contributed greatly to its popularity.
• Technological advancements and infrastructure improvements allowed for widespread availability and reliability of cable services.
• The year-on-year increase in cable television households reflects changes in lifestyle, as cable TV provided households with varied entertainment and an alternative to traditional broadcast networks.

The impressive growth rate that we calculated highlights the rapid expansion of cable television, reflecting how technological innovations can quickly become mainstream.

Household Statistics

Household statistics provide crucial insights into trends and changes in society. In this case, tracking the number of households with cable television from 1977 to 2003 gives valuable information about consumer behavior and the dissemination of new technology. Several aspects of household statistics are worth noting:

• These statistics enable us to understand penetration rates of new consumer technologies, like cable television.
• They help governments, businesses, and analysts make informed decisions based on how consumers adapt to and embrace technological tools.
• Studying historical data on households can reveal patterns that might help anticipate future trends in technology adoption.

In summary, data on household statistics are indispensable tools for analyzing past trends and predicting future consumer technology usage.

Time Series Analysis

Time series analysis involves examining data points collected or recorded at successive points in time. It is essential in many fields, including economics, finance, and technology adoption. When analyzing the growth of cable television, time series analysis helps to identify trends, patterns, and rates of change over time. When applying time series analysis to cable television growth:

• We look at discrete data points, in this case, the number of households in specific years (1977 and 2003).
• The analysis shows how such data points can be connected to provide insights into average rates of change, an important metric for understanding growth.
• It involves calculating differences between observed values to assess how quickly or slowly a trend unfolds over a specified period.

Using techniques like these not only helps in understanding historical trends but also in forecasting future developments, making time series analysis a critical component of modern data analysis.