Problem 47 - 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 an average of 10 hours of talk shows per week?

Step-by-Step Solution

Answer

According to the model, a household will watch an average of 10 hours of talk shows per week in the year 2002.

Step 1: Substitute the value of \(h\) in the equation

We are told in the problem statement that we are looking for the year when \(h=10\). Thus, we substitute \(10\) for \(h\) in the equation, getting \(10=0.35t+4.06\).

Step 2: Solve the equation for \(t\)

In order to solve for \(t\), we isolate \(t\) on one side of the equation. To do this, we subtract \(4.06\) from both sides of the equation. This gives us \(10-4.06=0.35t\), or \(5.94=0.35t\). To isolate \(t\), we divide both sides of the equation by \(0.35\), giving us \(t=5.94/0.35\). This gives an approximate value of \(t\approx17\).

Step 3: Translate \(t\) into the year from 1985

Since \(t\) represents the number of years since 1985, we add \(t\) to 1985 to get the year. Therefore, \(1985+17=2002\).

Key Concepts

Solving Equations

When solving equations, the primary goal is to isolate the unknown variable. In this exercise, we are dealing with the equation related to the number of hours spent watching talk shows, which is given by \[ h = 0.35t + 4.06 \].To find out when an average household will watch 10 hours of talk shows per week, we substitute \( h = 10 \) into the equation, leading to \[ 10 = 0.35t + 4.06 \]. To solve for \( t \), we need to perform some operations to isolate it on one side of the equation.

• First, subtract 4.06 from both sides to move constants away from \( t \).
• This yields \[ 10 - 4.06 = 0.35t \], simplifying to \[ 5.94 = 0.35t \].
• Lastly, divide both sides by 0.35 to solve for \( t \).

This gives \( t \approx 17 \), representing the number of years since 1985 that it would take to watch the desired number of hours.

Mathematical Models

Mathematical models help represent real-world scenarios using algebraic expressions. Here, we have different equations modeling the U.S. household TV watching behavior across various types of shows. Each expression provides a linear relationship between time \( t \) and hours \( h \) spent watching.

• Model 1: General TV watching - \[ h = 0.57t + 54.85 \]
• Model 2: Watching talk shows - \[ h = 0.35t + 4.06 \]
• Model 3: Watching game shows - \[ h = -0.2t + 3.4 \]

The coefficients represent the change in hours over time, with positive values indicating an increase and negative values indicating a decrease. These models are simplified predictions, offering insight into trends over the years.

Understanding these models helps students grasp how mathematics can analyze trends and predict future behavior, based on assumptions about past data.

Year Calculation

Once we solve for \( t \) in the context of these models, it is important to translate that into an actual calendar year. In this exercise, \( t \) represents the number of years since 1985.

• Start with the year 1985, which serves as the initial time point (\( t = 0 \)).
• Add the value of \( t \) you've calculated to this base year.

For example, with \( t \approx 17 \), we calculate the year by adding 17 to 1985. This gives us \[ 1985 + 17 = 2002 \].

Translating \( t \) into a year is crucial for interpreting results in a meaningful way, allowing viewers to contextualize mathematical solutions for real-world use.