Question

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 . Gas spending model: \(N=1.488 t^{2}-3.403 t+65.590\) Electricity spending model: \(E=-0.107 t^{2}+6.897 t+169.735\) Find a model for the total amount \(A\) (in billions of dollars) spent on natural gas and electricity by United States residents from 1989 through 1993 . (GRAPH CAN NOT COPY)

Short answer

The model for the total amount (A) spent on natural gas and electricity by United States residents from 1989 to 1993 is given by \(A = 1.381 t^{2} + 3.494 t + 235.325\).

Step-by-step solution

Understand the Given Model Equations

The question provides two model equations, one for the money spent on natural gas (N) and one for electricity (E). Both are in terms of t (the number of years since 1989). The gas spending model is given as \(N=1.488 t^{2}-3.403 t+65.590\). The electricity spending model is given as \(E=-0.107 t^{2}+6.897 t+169.735\).

Form the Model for Total Amount Spent

To find the model for the total amount (A) spent, we can add the two model equations together. \(A = N + E\). Substitute the given equations for N and E into the formula.

Solve the Equation

On substituting the given values, we get \(A = (1.488 t^{2}-3.403 t+65.590) + (-0.107 t^{2}+6.897 t+169.735)\). The equation simplifies to \(A = (1.488 - 0.107) t^{2} + (-3.403+6.897) t + (65.590+169.735)\). This simplifies to \(A = 1.381 t^{2} + 3.494 t + 235.325\). This is the model for the total amount spent on natural gas and electricity by United States residents from 1989 through 1993.

Key concepts

Algebraic Model Equations

When we look at algebraic model equations, we are actually discussing the creation of mathematical representations that reflect real-world scenarios. In our case, the spending on natural gas and electricity over a period of time has been transformed into two separate algebraic equations. Here's what each term of these algebraic equations represents:

• The coefficient of the quadratic term, which is the term with the variable raised to the second power (for example, 1.488 in the gas model), indicates the rate of acceleration (increase or decrease) in spending over time.
• The coefficient of the linear term, with the variable to the first power (for instance, -3.403 in the gas model), signifies the initial rate of change in spending.
• The constant term (like 65.590 in the gas model) gives us the starting point of spending in the base year, which is 1989 in this scenario.

Constructing algebraic models like these allows one to predict future spending, analyze past trends, and understand various aspects of spending behavior. Their use is extensive in economics, engineering, and scientific research, among other fields.

Quadratic Functions

Quadratic functions are one of the essential building blocks of algebra. They are polynomial functions of degree two, which means the highest exponent of the variable is two. In a quadratic function like \(f(x) = ax^2 + bx + c\), 'a', 'b', and 'c' are constants, and 'x' represents the variable. The graph of a quadratic function is a curve called a 'parabola'. These parabolic shapes are symmetrical, and they open upwards if the coefficient 'a' is positive, and downwards if 'a' is negative.

In the context of our exercise, both spending on natural gas and electricity are modeled with quadratic functions. These functions allow for the consideration of changes over time, providing a visualization for trends such as increases or decreases in spending. This becomes very powerful for making predictions. The sum of two quadratic equations, as in this exercise, is also a quadratic function, which means the model for total spending still allows us to apply the useful properties of quadratic functions in analysis.

Systems of Equations

Systems of equations involve two or more equations working together at the same time. Each equation in a system can be thought of as describing a line, curve, plane or some other shape in a space, and the solution to a system is the point or set of points that all of these shapes have in common. In simple terms, a solution to a system of equations is the answer that makes all the equations true at the same time.

In our exercise, the system was limited to two equations representing gas and electricity spending. By adding them together, we can find a new equation that models the total spending. Solving these systems can often be done by methods such as substitution (as in this situation where we substituted N and E in the equation \(A = N + E\)), graphing, or elimination. Each method has its use-cases, but fundamentally, they all serve to find a consistent solution across all equations within the system.