Question

From 1990 to 2010, the numbers (in millions) of female F and male M employees from the ages of 16 to 19 in the United States can be modeled by \(F(t)=-0.007 t^2+0.10 t+3.7\) and \(M(t)=0.0001 t^3-0.009 t^2+0.11 t+3.7\), where \(t\) is the number of years since 1990 . a. Find \((F+M)(t)\). b. Explain what \((F+M)(t)\) represents.

Short answer

a) \((F+M)(t) = 0.0001t^3 -0.016t^2 + 0.21t + 7.4\).\nb) \((F+M)(t)\) represents the total number of employees, both female and male, aged between 16 and 19 in the U.S. for a specific year (\( t \) years since 1990).

Step-by-step solution

Understand the Problem and Functions

F(t) and M(t) represent the number of female and male employees (in millions) respectively, from ages of 16 to 19 in the U.S., based on the years since 1990. This means that by substituting a certain value for \( t \), the output is the corresponding number of employees in millions for that particular year.

Adding the Functions

To find \((F+M)(t)\), we have to add the function F(t) and function M(t). So, \((F+M)(t) = F(t) + M(t) = (-0.007t ^2 + 0.10t + 3.7) + (0.0001t^3 - 0.009t^2 + 0.11t + 3.7) = 0.0001t^3 -0.016t^2 + 0.21t + 7.4 \), where \( t \) is the number of years since 1990.

Interpret the Resulting Function

\((F+M)(t)\) represents the total number of employees (both female and male) aged between 16 and 19 in the U.S., with \( t \) signifying the number of years since 1990. For example, by substituting the value of \( t \) into \((F+M)(t)\), one can determine the total number of employees in a particular year.

Key concepts

Modeling Relationships

Modeling relationships in mathematics often involves creating equations to represent real-world scenarios, allowing us to understand and predict behavior over time. In this context, we are modeling the relationships between years and the number of employees aged 16 to 19 in the United States from 1990 to 2010.

Using polynomial functions, such as the ones given for female employees, \( F(t) = -0.007 t^2 + 0.10 t + 3.7 \), and male employees, \( M(t) = 0.0001 t^3 - 0.009 t^2 + 0.11 t + 3.7 \), allows us to capture trends and changes over this timeframe.

These equations incorporate different powers of \( t \), representing complexities in how these populations have changed. For instance, the presence of \( t^2 \) and \( t^3 \) terms suggest that these employee numbers don't change linearly but might have more nuanced fluctuations.

By modeling such relationships, we can analyze shifts in employment patterns and even foresee future trends based on historical data.

Function Addition

Function addition is a fundamental mathematical operation where two functions are combined. In our example, we add the functions \( F(t) \) and \( M(t) \) to find the total number of young employees, both female and male.

Let's break this down: to find \((F+M)(t)\), we add corresponding terms from each function. For instance,

• the \( t^3 \) term results solely from \( M(t) \) as \( F(t) \) lacks a \( t^3 \) term,
• the \( t^2 \) terms combine to \( -0.007t^2 - 0.009t^2 = -0.016t^2 \),
• the \( t \) terms add up to \( 0.10t + 0.11t = 0.21t \), and
• the constant term, \( 3.7 + 3.7 = 7.4 \).

By combining these, we derive a new polynomial \( 0.0001t^3 -0.016t^2 + 0.21t + 7.4 \). This new function summarizes the total number of young employees over the years, making it a useful tool for analysis.

Real-World Applications

Polynomial functions and operations like addition are not just abstract mathematical concepts; they have real-world applications, particularly in analyzing and forecasting demographics and economics.

In our example, these functions can offer insights into social changes, such as shifts in workforce demographics among young people in the United States. Policymakers can use this information to inform decisions about education and youth employment programs.

Moreover, businesses might utilize such data to align their recruitment strategies with changing trends, while educators could tailor programs to better prepare young people for employment opportunities.

Overall, understanding and utilizing polynomial functions to model real-world data helps us make more informed decisions across a wide range of fields, reflecting how deeply interwoven mathematics is with our daily lives.