Problem 54 - Algebra 1: Concepts and Skills

Question

In Exercises 53 and \(54,\) use the following information. From 1989 through \(1993,\) the amounts (in billions of dollars) spent on natural gas \(N\) and electricity \(E\) by United States residents can be modeled by the following equations, where \(t\) is the number of years since \(1989 .\) $$\text {Gas spending model:} N=1.488 t^{2}-3.403 t+65.590$$ $$\text {Electricity spending model:} E=-0.107 t^{2}+6.897 t+169.735$$ According to the models, will more money be spent on natural gas or on electricity in \(2020 .\) HINT: It may be helpful to graph the equations on a graphing calculator to answer this question.

Step-by-Step Solution

Answer

According to the models, more money will be spent on natural gas than on electricity in 2020.

Step 1: Determine the value of the variable \(t\)

Since the models start from 1989, the value of \(t\) for the year 2020 would be \(2020 - 1989 = 31.\)

Step 2: Substitute the value of \(t\) into the Gas spending model

According to the given model, \(N=1.488 t^{2}-3.403 t+65.590\). Substitute \(t=31\) into the equation we get: \(N = 1.488 * (31)^2 - 3.403 * 31 + 65.59 = 1484.18\) billion dollars.

Step 3: Substitute the value of \(t\) into the Electricity spending model

Likewise, by substituting \(t=31\) into the equation \(E=-0.107 t^{2}+6.897 t+169.735\), we get: \(E = -0.107 * (31)^2 + 6.897*31 + 169.73 = 598.90\) billion dollars.

Step 4: Compare the spending on Gas and Electricity

By comparing, we can observe that \(N > E\), which implies that higher spending is predicted in the case of Natural gas than on Electricity in 2020.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in understanding many natural and economic phenomena. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. This standard form is often used to model relationships where changes do not happen linearly. Instead, they might reflect accelerating, decelerating, or other nonlinear patterns. In the context of the problem, quadratic equations are used to model the spending on natural gas and electricity, allowing us to predict future spending based on past data. These models are beneficial because they provide a mathematical framework to forecast trends over time by substituting different values of \( t \) to project future outcomes.

Graphing Calculator

A graphing calculator is an essential tool in visualizing complex functions like quadratic equations. By displaying function graphs, it helps us understand and analyze mathematical relationships quickly. When dealing with models that involve quadratic equations, plotting them on a graphing calculator will show their parabolic nature. In our exercise, students are encouraged to use a graphing calculator to graph the provided equations for natural gas and electricity spending against time.

• This graphical representation helps in visualizing when one expenditure surpasses the other.
• It further aids in a visual comparison for the specified year, enhancing comprehension beyond numerical calculations alone.

Using this device, one can predict critical points or intersections that can inform decisions or predictions such as which resource will have a higher expenditure in 2020.

Yearly Predictions

Yearly predictions enable us to estimate future occurrences based on historical data. In mathematical modeling, these predictions allow for informed decision making in economics, science, and social sciences.

• In the current problem, the equations model spending over years and enable predictions for 2020 using a specific value of \( t \).
• This process involves determining what past trends might tell us about future events, leveraging the nature of quadratic equations.

By plugging the value of \( t = 31 \) into our models, where \( t \) represents years since 1989, we calculated anticipated spending on natural gas and electricity. These results show that there would be greater expenditure on natural gas compared to electricity in 2020. This illustrates the usefulness of yearly predictions in policy planning and business forecasting.

Substitution Method

Substitution is a straightforward technique employed in mathematical modeling to find specific values for unknowns in equations. It involves replacing a variable with a specific number or expression. In our exercise,

• The substitution method is used to assess expenditure on fuel sources by substituting the value of \( t = 31 \) into both quadratic models.
• For the gas spending model \( N = 1.488 t^2 - 3.403 t + 65.590 \), substitution helped find that natural gas spending would be approximately \(1484.18 billion in 2020.
• Similarly, for electricity \( E = -0.107 t^2 + 6.897 t + 169.735 \), the substitution gave us a spend of \)598.90 billion.

Through substitution, we not only simplify the problem but also derive concrete predictions from abstract models, making these predictions feasible and actionable.