Problem 46 - Algebra 1

Question

They can be used to estimate the average number of hours \(h\) per week that a household in the United States watches television programs. In all three models, \(t\) is the number of years since \(1985 .\) Hours spent watching television: \(h=0.57 t+54.85\) Hours spent watching talk shows: \(\quad h=0.35 t+4.06\) Hours spent watching game shows: \(h=-0.2 t+3.4\) According to the model, in what year will a household watch television an average of 70 hours per week?

Step-by-Step Solution

Answer

According to the model, a household will watch television an average of 70 hours per week in the year 2012.

Step 1: Model Equation Identification

Identify the equation for total television viewing habits. It is given by \(h=0.57t+54.85\).

Step 2: Set the Total Viewing Time to 70 Hours

Set the equation \(h=0.57t+54.85\) to 70 hours and solve for \(t\). This gives us \(0.57t = 70 - 54.85\), which simplifies to \(0.57t = 15.15\).

Step 3: Calculate the Number of Years

To calculate the number of years since 1985, divide both sides of the equation by 0.57. This gives us \(t = 15.15 / 0.57\), which is approximately \(26.58\).

Step 4: Find the Year

The value of \(t\) represents the number of years since 1985. So we add 1985 to \(t\), and we get the year as \(1985 + 26.58 = 2011.58\). Since we cannot have a decimal value for a year, we have to round this to the nearest whole number. This gives us the year as 2012.

Key Concepts

Time Calculation

Time calculation in linear equations is a fascinating concept that allows us to predict outcomes over time. When we have a linear equation such as \( h = 0.57t + 54.85 \), which represents hours of television watched as a function of years since 1985, we can use it to find out specific time points.

• The variable \( t \) here represents the number of years after 1985. For example, if \( t \) is 10, it signifies the year 1995.
• To find out when a certain condition is met—in this case, when the average viewing hours reach 70—we substitute \( h = 70 \) into the equation.
• Solving for \( t \) gives us the number of years after 1985 when this condition occurs. This is an essential skill in analyzing trends and making future predictions.

Such calculations are vital in fields like economics, epidemiology, and environmental science where forecasting future trends is crucial.

Modeling Real-World Data

Modeling real-world data using linear equations helps in understanding and interpreting several phenomena effectively. Linear models like \( h = 0.57t + 54.85 \) simplify complex systems by assuming a constant rate of change.

• In our example, the equation expresses a direct relationship between time and television hours: each year adds 0.57 hours to the weekly average.
• Models can be used for descriptive purposes to see past patterns, or predictively to estimate future values.
• Assumptions are key in every model. In linear modeling, we assume that changes are uniform, meaning every unit of increase in \( t \) changes \( h \) by 0.57 consistently.

Understanding these models can help in making well-informed decisions, whether in business forecasts, policy-making, or personal planning.

Estimation Techniques

Estimation techniques in linear modeling are essential for solving real-world problems where precise values are not always available.

• Our example demonstrates this: we cannot have a fractional year, so \( t = 26.58 \) needs to be estimated to the closest whole number.
• In mathematics, this is translated to rounding. Since \( 26.58 \) is closer to 27 than to 26, we round it to 27.
• This whole number helps us pinpoint an actual calendar year, giving us a practical, understandable context—here, indicating the year 2012.

Estimation is widely used in budget planning, time management, and even scientific calculations where measurements often come with a degree of uncertainty.