Question

MAKE A DECISION The enrollment \(E\) in an early childhood development program for a school district from 1995 to 2008 can be approximated by the model \(E=1.678 t^{2}+1025,5 \leq t \leq 18\), where \(t\) represents the year, with \(t=5\) corresponding to \(1995 .\) Use the model to approximate the year in which the early childhood enrollment reached 1450 children. Can you use the model to estimate early childhood enrollment for the year 1980 ? Explain.

Short answer

The model approximates that the enrollment in the early childhood development program reached 1450 children around the year 2000. However, this model cannot be used to estimate the early childhood enrollment for the year 1980, since that year falls outside the established valid domain for the model.

Step-by-step solution

Determine the year when enrollment is 1450

First, to find out the year when the enrollment reached 1450 students, we need to set \(E\) equal to 1450 in the given model and solve for \(t\). \nThis can be done by solving the quadratic equation \(1.678t^2 + 1025 = 1450\).

Solve for \(t\)

Now solve for \(t\). Rearrange the equation to find the zero of the quadratic equation: \(1.678t^{2} + 1025 = 1450\) becomes \(1.678t^{2} = 1450 - 1025\). Solving for \(t\), we get two roots, only one of which falls within the given domain of \(t\). Choose the valid root for further calculation.

Use \(t\) to find the year

The value of \(t\) calculated corresponds to the number of years after 1990. Adding this number to 1990 gives the year for which the enrollment was 1450. This is the desired approximate year for which the early childhood enrollment reached 1450 children.

Evaluate the validity of the model for the year 1980

To answer whether we can use the model to estimate early childhood enrollment for the year 1980, we need to consider the domain given. According to the model, \(t\) falls within the [5, 18] interval, representing the years from 1995 to 2008. As 1980 falls outside this range, we can conclude that the model is not valid for predicting the enrollment in 1980.

Key concepts

Early Childhood Enrollment

Understanding early childhood enrollment trends is important for school districts as they plan for resources and staffing. Early childhood programs aim to prepare children for elementary school by developing cognitive, emotional, and social skills. Monitoring enrollment numbers over time helps districts identify growth patterns and allocate the necessary support. In the given problem, we evaluate the enrollment trends from 1995 to 2008 using a mathematical model, which provides an approximation of the number of children attending these programs during those years. This data-driven approach assists in anticipating future needs and ensuring that quality education is accessible to all young learners.

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). In this context, the quadratic equation is used to model the number of children enrolled in early childhood programs over specific years. For the given model, the equation is \(1.678t^2 + 1025 = E\). The variable \(t\) represents time in years since 1990, making it essential for finding the exact year when the enrollment reached a particular number, such as 1450 children. Solving a quadratic equation often involves finding the roots, which are the values of \(t\) that satisfy the equation. In practical terms, this means determining the year when the enrollment reached a specific milestone.

Domain and Range

In mathematics, the domain refers to all the possible input values that a function can accept, while the range refers to all the possible outputs. In the enrollment model provided, the domain is specified as \(5 \leq t \leq 18\). This means the model is valid for years 1995 through 2008. The range represents all potential enrollment numbers that can be forecasted within this period. When addressing whether the model can estimate enrollment for 1980, it’s crucial to recognize that 1980 falls outside the specified domain. As such, the model's predictions are unreliable for that year because it extrapolates beyond the intended data range, potentially leading to inaccurate results.

Approximation in Algebra

In algebra, approximation is used when exact solutions are either impossible or unnecessary. Quadratic modeling, such as estimating early childhood enrollment, often uses approximations to make predictions meaningful. By plotting real-world data, mathematicians derive equations that approximate trends. However, these models are not perfect; they serve as close estimates based on observed data within a specified time range or domain. The accuracy depends on the validity of the model within the defined boundaries. When extrapolated to years or situations outside these parameters, as seen with the 1980 estimation attempt, the approximations become less reliable. Despite this, approximations provide valuable insights and predictions, helping educators and planners make informed decisions based on historical trends.