Problem 41 - Algebra 1

Question

Use the following information. The per capita consumption (pounds per person) of whole milk W and reduced fat milk R from 1980 to 1995 can be modeled by the equations below where t represents years since 1980. $$\begin{aligned}&\text { Whole milk: } W=0.09 t^{2}-5.29 t+114.69\\\&\text { Reduced fat: } R=-0.23 t^{2}+2.40 t+71.39\end{aligned}$$ Sketch a graph of each equation on the same coordinate plane.

Step-by-Step Solution

Answer

When you plot these equations on the same coordinate plane, you see a downward-opening parabola for reduced fat milk and an upward-opening parabola for whole milk. The specific location and shape of these parabolas depends on their specific equations. An intersection point would indicate a trend change in milk consumption.

Step 1: Identify the quadratic equations

The two quadratic equations are given as:\n For Whole milk: \(W = 0.09t^{2} - 5.29t + 114.69\)\n For Reduced fat: \(R = -0.23t^{2} + 2.40t + 71.39\) where \(t\) represents years since 1980.

Step 2: Plot the equations

To sketch these equations, you can start by identifying the vertex of each parabola. The vertex form of a quadratic function is \(y=a(x-h)^{2}+k\) where \((h,k)\) is the vertex. You can convert the given equations into this form or you may find the vertex by using the formula \(h=-b/(2a)\), where \(a\) and \(b\) are coefficients in the quadratic equation \(y = ax^{2} + bx + c\). Once you've identified the vertex, you can find a few more points by plugging in values for \(t\) on either side of the vertex and then sketch the parabolas. For the whole milk equation, since \(a > 0\) the parabola will face upwards and for reduced fat equation since \(a < 0\), the parabola will face downward.

Step 3: Interpret the graphs

The graphs represent changes in per capita consumption of whole milk and reduced fat milk over time. Where the graphs intersect, it signifies a change in trend. For instance, if the reduced fat milk graph lies below the whole milk graph at first and then crosses it to lie above, it means that initially more whole milk was consumed than reduced fat milk, but after the point of intersection, more reduced fat milk started to be consumed.

Key Concepts

Graphing Parabolas

Graphing parabolas is a foundational skill in algebra which involves plotting the curved pathway of quadratic functions on a coordinate plane. The general form of a quadratic equation is \( y = ax^2 + bx + c \). The shape of the parabola is determined by the value of the coefficient \(a\): if \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.

When graphing parabolas for the given milk consumption equations, the first step is to identify the direction in which each parabola opens based on the coefficient of \(t^2\). For whole milk, the equation \( W=0.09t^2-5.29t+114.69 \) has a positive \(a\), indicating an upward opening parabola. Conversely, the reduced fat milk equation \( R=-0.23t^2+2.40t+71.39 \) features a negative \(a\), pointing to a downward opening parabola.

To create an accurate graph, select several values of \(t\), calculate the corresponding \(W\) and \(R\), and plot these points on the plane. A smooth curve through these points will give you the parabolas for whole and reduced fat milk consumption. Observing the intersection and behavior of these parabolas provides insights into consumption trends over time.

Vertex of a Parabola

The vertex of a parabola is a crucial point representing the maximum or minimum value of the quadratic function, which is significant in analyzing and graphing parabolas. For the equations representing milk consumption, determining the vertex allows us to understand the peak years for whole milk or reduced fat milk consumption.

To find the vertex, you can use the formula \(h=-\frac{b}{2a}\), where \(h\) is the x-coordinate of the vertex and \(a\) and \(b\) are coefficients from the equation \(y=ax^2+bx+c\). After finding \(h\), substitute it back into the original equation to get the y-coordinate, represented by \(k\).

For example, the vertex of the whole milk consumption equation \(W=0.09t^2-5.29t+114.69\) can be calculated using the coefficients from the equation, resulting in a vertex that shows the year and the amount of peak whole milk consumption. Understanding the vertex helps in interpreting the data and knowing at what point one form of milk surpassed the other in popularity.

Per Capita Consumption Analysis

Per capita consumption analysis is an important method in economics and health sciences for understanding the average consumption behavior in a particular population. In the context of our exercise, graphs of whole and reduced fat milk consumption can reveal trends and shifts in dietary preferences.

By graphing the quadratic equations of milk consumption over time, analysts can observe changes in consumer behavior, such as a shift from whole milk to reduced fat milk. The per capita consumption is modeled as a function of time, and the curves illustrate increases or decreases in average consumption.

Interpreting these graphs allows for an analysis of dietary trends, and when combined with data on health outcomes or market changes, it can provide comprehensive insights into public health and economic dynamics of the dairy industry. Notably, where the parabolas intersect indicates a significant change in consumption habits, marking a point where either whole milk or reduced fat milk became more popular.