Problem 92
Question
The median age of women at first marriage is approximated by $$ A(t)=0.08 t+19.7 $$ where \(A(t)\) is the median age of women marrying for the first time at \(t\) years after \(1950 .\) a) Find the rate of change of the median age \(A\) with respect to time \(t\) b) Explain the meaning of your answer to part (a).
Step-by-Step Solution
Verified Answer
a) The rate of change is 0.08. b) This indicates the median age at first marriage increases by 0.08 years each year after 1950.
1Step 1: Understand the Function
The function given is \( A(t) = 0.08t + 19.7 \). This function models the median age \( A \) of women at first marriage \( t \) years after 1950. \( A(t) \) is a linear function in terms of \( t \), where \( 0.08 \) is the coefficient of \( t \) and \( 19.7 \) is the y-intercept.
2Step 2: Determine the Rate of Change
In a linear function \( A(t) = mt + b \), the coefficient \( m \) represents the rate of change. Here, the coefficient of \( t \) is \( 0.08 \). Thus, the rate of change of \( A \) with respect to \( t \) is \( 0.08 \).
3Step 3: Interpret the Rate of Change
The rate of change, \( 0.08 \), indicates how much the median age \( A \) of first-time marrying women increases per year after 1950. It means that each year, the median age at first marriage increases by 0.08 years.
Key Concepts
Rate of ChangeLinear FunctionMathematical Modeling
Rate of Change
In mathematics, the 'rate of change' is a measure that describes how one quantity changes in relation to another. It's a fundamental concept in calculus and is often represented by the slope of a line in linear functions or as a derivative in more complex functions.
For linear functions, the rate of change is constant. In the given exercise, the function is linear, meaning the rate at which the median age changes over time remains the same each year.
In practical terms, understanding the rate of change can provide insights into trends and predictions. For this exercise, the rate of 0.08 represents the annual increase in the median age since 1950. It's a small but steady change, indicating a gradual shift in societal norms.
For linear functions, the rate of change is constant. In the given exercise, the function is linear, meaning the rate at which the median age changes over time remains the same each year.
In practical terms, understanding the rate of change can provide insights into trends and predictions. For this exercise, the rate of 0.08 represents the annual increase in the median age since 1950. It's a small but steady change, indicating a gradual shift in societal norms.
Linear Function
A linear function is a type of function that creates a straight line when graphed. It has the form \( f(x) = mx + b \), where \( m \) is the slope (or rate of change) and \( b \) is the y-intercept. Linear functions are simple yet powerful tools in mathematical modeling.
In the context of the given problem, the function \( A(t) = 0.08t + 19.7 \) perfectly fits the definition. Here, \( 0.08 \) is the slope, showing how quickly or slowly the median age increases each year. The y-intercept \( 19.7 \) represents the median age at the base year, i.e., 1950, when the time \( t = 0 \).
Recognizing a problem as a linear function helps in predicting future values through simple calculations, making it invaluable in fields like economics, physical sciences, and everyday decision-making.
In the context of the given problem, the function \( A(t) = 0.08t + 19.7 \) perfectly fits the definition. Here, \( 0.08 \) is the slope, showing how quickly or slowly the median age increases each year. The y-intercept \( 19.7 \) represents the median age at the base year, i.e., 1950, when the time \( t = 0 \).
Recognizing a problem as a linear function helps in predicting future values through simple calculations, making it invaluable in fields like economics, physical sciences, and everyday decision-making.
Mathematical Modeling
Mathematical modeling involves using mathematical equations and functions to represent real-world situations. It's an essential part of various fields, including engineering, economics, and science.
By creating a model, we can simulate complex behaviors and predict future outcomes based on current data. In this particular case, the function \( A(t) = 0.08t + 19.7 \) models the median age of women at first marriage over years.
This model is particularly useful because it simplifies a complex trend into a format that's easy to understand and manipulate. While real-life data might have fluctuations, a linear model provides a generalized overview that can still offer substantial insights and assist in making informed decisions. It's a snapshot of historical trends and can be the foundation for further, more detailed analysis as more data becomes available or as different factors are introduced.
By creating a model, we can simulate complex behaviors and predict future outcomes based on current data. In this particular case, the function \( A(t) = 0.08t + 19.7 \) models the median age of women at first marriage over years.
This model is particularly useful because it simplifies a complex trend into a format that's easy to understand and manipulate. While real-life data might have fluctuations, a linear model provides a generalized overview that can still offer substantial insights and assist in making informed decisions. It's a snapshot of historical trends and can be the foundation for further, more detailed analysis as more data becomes available or as different factors are introduced.
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Problem 91
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