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. $$\begin{aligned}&\text { Whole milk: } W=0.09 t^{2}-5.29 t+114.69\\\&\text { Reduced fat: } R=-0.23 t^{2}+2.40 t+71.39\end{aligned}$$ Find the points where the parabolas intersect. What does each of these points represent?

Short answer

The points of intersection are the solutions to the quadratic equation 0.32t^2 - 7.69t + 43.3 = 0. They represent the years since 1980 (t) when the consumption of whole and reduced fat milk are the same.

Step-by-step solution

Setting Equations Equal To Each Other

Since we need to find when the consumption of whole milk equals the consumption of reduced fat milk, we must set the two equations equal to each other. This gives us the new equation:0.09t^2 - 5.29t + 114.69 = -0.23t^2 + 2.4t + 71.39.

Solve for t

By bringing all terms to one side of the equation, we have:0.09t^2 + 0.23t^2 - 5.29t - 2.4t + 114.69 - 71.39 = 0. Simplify this to:0.32t^2 - 7.69t + 43.3 = 0Now, solve this quadratic equation for \(t\) using the quadratic formula \(t = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}\), where \(a= 0.32\), \(b= -7.69\) and \(c= 43.3\).

Substitute t and Solve for W or R

Substitute the solving values of \(t\) gotten from step 2 into either the \(W\) or \(R\) equation to find the corresponding consumption.

Interpret the Results

The \(t\) values represent the years since 1980 and the \(W\) or \(R\) value represents the per capita milk consumption in those years. These data points represent when the consumption of whole milk is the same as the consumption of reduced fat milk.

Key concepts

Parabolas Intersection

When working with equations that model real-world situations, an interesting event occurs when two parabolas intersect. This indicates that the two scenarios represented by each parabola have the same value at the point of intersection.

In the context of the given problem, the parabolas represent the consumption of whole milk and reduced fat milk over time. Finding their points of intersection translates to discovering exact moments in time when both consumptions are identical.

To determine this, we align the equations of both parabolas by setting them equal to each other - essentially finding the common solution for both functions. Once the points are found, analyzing these moments can give insights into trends and preferences in milk consumption over the studied period.

Solving Quadratic Equations

Quadratic equations are fundamental in algebra and come in the form of \(ax^2 + bx + c = 0\). The solutions to these equations are where the parabola, which is the graph of a quadratic function, crosses the x-axis.

To solve a quadratic equation, you can use various methods, such as factoring, completing the square, or employing the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In the exercise, the quadratic formula is the most effective approach because it provides a systematic way to find the real solutions regardless of whether the equation can be factored easily.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world phenomena. The model uses equations to mimic how certain variables—like time or consumption in our exercise—interact with one another.

Models can be simple linear relationships or complex functions like the quadratic equations used to represent milk consumption in this case. The accuracy of these models is paramount, as they can inform policy-making, economic forecasting, and in our example, provide insights into dietary trends.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations (such as addition and multiplication). These expressions are the building blocks of algebra and are used to represent relationships between different quantities.

In our milk consumption problem, the algebraic expressions \(W = 0.09t^2 - 5.29t + 114.69\) and \(R = -0.23t^2 + 2.40t + 71.39\) encapsulate complex information about milk consumption dynamics in a concise manner. By manipulating these expressions according to algebraic laws, we can explore and solve for unknown values that give them real-world significance.