Problem 33 - College Algebra with Modeling and Visualization

Question

Use the midpoint formula for the following. The average life expectancy for a female born in the United States was 77.4 years in 1980 and 79.5 years in 2000 . Estimate the average life expectancy for a female born in 1990 . (Actual life expectancy was \(78.8 .\) ) (Source: Bureau of the Census.)

Step-by-Step Solution

Answer

The estimated average life expectancy for a female born in 1990 is 78.45 years.

Step 1: Understanding the Midpoint Formula

The midpoint formula, used when discussing averages or rates that change over time, is \( M = \frac{(x_1+y_1) + (x_2+y_2)}{2} \). In this case, we will apply it to life expectancy data to find the estimated life expectancy in the midpoint year, 1990.

Step 2: Identifying Given Values

From the problem, we have the life expectancy in 1980 as 77.4 years and in 2000 as 79.5 years. These will be our \( y_1 \) and \( y_2 \), respectively. The years 1980 and 2000 serve as \( x_1 \) and \( x_2 \), but for the midpoint estimation, they are not directly used in calculation.

Step 3: Applying the Midpoint Formula

To find the estimated life expectancy for 1990, substitute the values into the midpoint formula for \( y \): \[ M = \frac{77.4 + 79.5}{2} \]. Calculate the sum first, which gives \( 156.9 \), then divide by 2.

Step 4: Calculating the Midpoint

Perform the division: \( \frac{156.9}{2} = 78.45 \). Thus, the estimated average life expectancy for a female born in 1990 is 78.45 years.

Key Concepts

Life Expectancy Estimation

Life expectancy estimation is a method used to predict the average lifespan of individuals born in a certain year. It uses historical data to make informed projections about how long a person is expected to live. This is particularly valuable for studies in public health and economics, where it helps in planning for future healthcare needs and social services. To estimate life expectancy, different mathematical tools and models can be used, including the midpoint formula, which provides a straightforward approach to finding averages over specific periods. The process involves:

• Collecting life expectancy data from different years.
• Applying suitable mathematical methods to estimate the life span for other years within that range.

In our example, we used the midpoint formula to estimate life expectancy for a year not explicitly given in the data, demonstrating this method's practical application.

Average Rate of Change

The average rate of change describes how much a quantity changes on average over a certain interval. This concept is crucial in various fields because it helps in understanding trends and patterns over time. To calculate the average rate of change:

• Identify two points on a graph, with each point having an associated time and value, such as year and life expectancy.
• Use the formula \( \text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1} \), where \( y_2 \) and \( y_1 \) are the life expectancies at different years \( x_2 \) and \( x_1 \).

In the exercise example, while the midpoint formula estimates the expectancy, the average rate of change would provide insight into how quickly life expectancy was improving from 1980 to 2000.

Mathematical Modeling

Mathematical modeling is the process of using mathematics to represent, analyze, and predict real-world phenomena. In life expectancy estimation, models can capture historical trends and project future values. The models help decision-makers understand potential outcomes and make informed choices. Key aspects of mathematical modeling include:

• Selecting the right type of model—linear, nonlinear, etc.—based on the data characteristics and the question at hand.
• Using equations and algorithms, like the midpoint formula, that simplify real-world data into comprehensible predictions.
• Continuously refining models as new data becomes available to improve accuracy.

In our estimation problem, the model used was simple, yet it provided valuable insights into the middle data point not originally given.