Problem 11 - Beginning and Intermediate Algebra

Question

Solve. The number of babies born to teenage mothers from 1989 to 2002 can be approximated by $$N(t)=-0.721 t^{2}+2.75 t+528$$ where \(t\) represents the number of years after 1989 and \(N(t)\) represents the number of babies born (in thousands). According to this model, in what year was the number of babies born to teen mothers the greatest? How many babies were born that year?

Step-by-Step Solution

Answer

The greatest number of babies born to teen mothers occurred in 1991, with approximately 531,836 babies born.

Step 1: Identify the quadratic function and its coefficients

We are given the quadratic function representing the number of babies born to teenage mothers: $$N(t) = -0.721t^2 + 2.75t + 528$$ The coefficients are: \(a = -0.721\), \(b = 2.75\), and \(c = 528\).

Step 2: Use the vertex formula to find the maximum value of t

The vertex formula is: $$t_{vertex} = \frac{-b}{2a}$$ Substitute the values of \(a\) and \(b\) from our quadratic function: $$t_{vertex} = \frac{-2.75}{2(-0.721)}$$ Calculate the value of \(t_{vertex}\): $$t_{vertex} \approx 1.908$$

Step 3: Substitute the vertex value of t into the function to find the maximum number of babies born

Next, we substitute the value of \(t_{vertex}\) into our original function, \(N(t)\), to find the maximum number of babies born: $$N(1.908) = -0.721(1.908)^2 + 2.75(1.908) + 528$$ Calculate the value of \(N(1.908)\): $$N(1.908) \approx 531.836$$

Step 4: Find the year when the maximum number of babies were born

We know that \(t\) represents the number of years after 1989. So, we can calculate the year when the maximum number of babies were born: $$1989 + t_{vertex} = 1989 + 1.908$$ Since we cannot have a fraction of a year, we round \(t_{vertex}\) to the nearest whole number: $$1989 + 2 = 1991$$

Step 5: Report the answer

According to this model, the greatest number of babies born to teen mothers occurred in 1991, with approximately 531,836 babies born (assuming we round the number to the nearest whole baby).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a critical point for understanding quadratic functions. It represents either the highest or lowest point of the graph, based on the direction in which the parabola opens. In the given exercise, the parabola opens downward because the coefficient of the square term (\(-0.721\)) is negative. This means the vertex provides the maximum point of the curve.

To find the vertex, we use the vertex formula:

• \[t_{\text{vertex}} = \frac{-b}{2a}\]

For the function \(N(t) = -0.721t^2 + 2.75t + 528\), substituting \(a = -0.721\) and \(b = 2.75\) gives:

• \[t_{\text{vertex}} = \frac{-2.75}{2(-0.721)} \approx 1.908\]

This calculation tells us when the maximum occurs in terms of \(t\), which is roughly 1.908 years after 1989.

Understanding the vertex helps in locating these key points for modeling real-world situations accurately.

Maximum Value of a Quadratic Equation

The maximum value of a quadratic equation occurs at the vertex for downward-opening parabolas. In our quadratic function, the maximum value is crucial because it signifies the greatest number of babies born to teenage mothers over the specified time period.

To find this maximum value, substitute \(t_{\text{vertex}}\) back into the original equation \(N(t) = -0.721t^2 + 2.75t + 528\):

• \[N(1.908) = -0.721(1.908)^2 + 2.75(1.908) + 528 \approx 531.836\]

This shows that, according to the model, approximately 531,836 babies were born in the peak year. The model allows us to understand the significance of the function's maximum point in a real-world context, like analyzing demographic trends.

Modeling with Quadratic Equations

Modeling with quadratic equations is essential in many fields like statistics, physics, and economics to predict and analyze trends. The given problem uses a quadratic equation to approximate the number of babies born to teenage mothers over several years.

When modeling such data:

• Choose an appropriate function type, here a quadratic, because it allows for capturing the rise and fall of data points.
• Identify key parameters and coefficients as they influence the shape and position of the parabola.
• Find significant points like the vertex to understand the behavior, such as maximum or minimum values.

The vertex in the problem gives insight into the year when births were highest, demonstrating how these equations can effectively model complex real-world phenomena. Such models can then be used by policymakers, educators, and researchers to inform decisions and strategies.