Problem 36 - College Algebra with Modeling and Visualization

Question

The median age of the U.S. population for each year \(t\) between 1970 and 2010 can be approximated by the formula \(A(t)=0.243 t-450.8\). (a) Compute the median ages in 1980 and 2000 . (b) What is the slope of the graph of \(A ?\) Interpret the slope.

Step-by-Step Solution

Answer

(a) 1980: 29.74 years, 2000: 35.2 years; (b) Slope is 0.243, meaning median age increases by 0.243 years per year.

Step 1: Understand the Formula

The function given is \( A(t) = 0.243t - 450.8 \), which expresses the median age of the U.S. population based on the year \( t \). Here, \( t \) is the year, and \( A(t) \) is the median age.

Step 2: Calculate Median Age for 1980

Substitute \( t = 1980 \) into the expression for \( A(t) \). So, \( A(1980) = 0.243 \times 1980 - 450.8 \). Simplifying gives \[ A(1980) = 480.54 - 450.8 = 29.74 \]. Hence, the median age in 1980 was approximately 29.74 years.

Step 3: Calculate Median Age for 2000

Substitute \( t = 2000 \) into the formula: \( A(2000) = 0.243 \times 2000 - 450.8 \). Simplifying gives \[ A(2000) = 486 - 450.8 = 35.2 \]. Therefore, the median age in the year 2000 was approximately 35.2 years.

Step 4: Determine the Slope of the Function

The slope of the function \( A(t) \) is the coefficient of \( t \), which is 0.243. This represents the average change in the median age for each passing year. Essentially, for every year that passes, the median age increases by 0.243 years.

Key Concepts

Slope Interpretation

The slope of a linear function like the one given in the exercise is very important to understand. It tells us how steep the line is when represented graphically and how much change occurs in the dependent variable for a unit change in the independent variable.

In the formula provided, the slope is the coefficient of the variable \(t\). Here, that coefficient is 0.243. This means that for each year that passes, the median age of the U.S. population increases by approximately 0.243 years.

Let's imagine this in everyday terms:

• If you were to plot this in a graph, each step you take horizontally to the right (representing one year) leads to a small climb upwards of 0.243 units in the vertical direction (representing the median age).
• This steady incline tells us that there’s a consistent increment in the median age over the years studied (from 1970 to 2010).

A consistent slope like this is helpful for predicting and understanding trends over time. It reflects consistent and gradual changes instead of abrupt shifts.

Median Age Calculation

The median age calculation using the formula involves simple substitution and arithmetic, but understanding the concept is crucial.

Median age is the middle point of a data set; half of the population is younger than this age, and half is older. To predict this number for a given year using the formula \(A(t) = 0.243t - 450.8\), you insert the desired year for \(t\). For example:

• For the year 1980, substitute \(t = 1980\).
• Calculate: \(A(1980) = 0.243 \times 1980 - 450.8 = 29.74\).

This tells us the median age in 1980 was approximately 29.74 years.

In the year 2000, the same process follows:

• Substitute \(t = 2000\) into the formula.
• Calculate: \(A(2000) = 0.243 \times 2000 - 450.8 = 35.2\).

Thus, the median age in 2000 was approximately 35.2 years.

These calculations show how the median age can be estimated for various years within the specified range using a straightforward method.

Linear Functions

Linear functions are one of the foundation stones of algebra and are characterized by their constant rate of change. The function given, \(A(t) = 0.243t - 450.8\), is linear because it fits the basic \(y = mx + c\) form, where \(m\) is the slope and \(c\) is the y-intercept.

Understanding linear functions helps consider various real-world relationships between two quantities that change together.

• The linear nature of the formula means any change in \(t\) directly affects the calculated \(A(t)\), with the change being constant, as dictated by the slope.
• The y-intercept \(-450.8\) represents the point where the line crosses the y-axis, or the projected median age at year zero — even if it doesn't have a practical real-world representation within this context, due to negative age values.

Linear functions like this become tools to predict and analyze changes over time, especially within a specified domain, like the population's median age between 1970 and 2010. They provide a simple yet powerful way to model trends and make approximate predictions.