The numbers of women y (in millions) in the civilian work...

The numbers of women $y$ (in millions) in the civilian work force in the United States from 2000 through 2010 (see figure) can be approximated by the model $y=0.63 t+66.3,0 \leq t \leq 10,$ where $t$ represents the year, with $t=0$ corresponding to 2000 (Source: U.S. Bureau of Labor Statistics) (GRAPH NOT COPY) (a) Find algebraically and interpret the $y$ -intercept of the graph. (b) According to the model, during which year did the number reach 72 million? (c) Explain how you can solve part (b) graphically and algebraically.

The numbers of women $y$ (in millions) in the civilian work force in the United States from 2000 through 2010 (see figure) can be approximated by the model $y=0.63 t+66.3,0 \leq t \leq 10,$ where $t$ represents the year, with $t=0$ corresponding to 2000 (Source: U.S. Bureau of Labor Statistics)
(GRAPH NOT COPY)
(a) Find algebraically and interpret the $y$ -intercept of the graph.
(b) According to the model, during which year did the number reach 72 million?
(c) Explain how you can solve part (b) graphically and algebraically.

Key Concepts

Y-intercept Interpretation

The y-intercept in a linear equation represents the value of the dependent variable when the independent variable is zero. It provides an initial condition or starting point for the modeled situation, allowing one to understand what the outcome would be at the beginning of the interval being considered.

Graphical Versus Algebraic Methods

Graphical and algebraic methods are two approaches to solving mathematical problems. The algebraic method involves using formulas and operations to find exact solutions, while the graphical method uses plots to visually identify intersections or trends that correspond to solutions. Both approaches are valuable and can sometimes complement each other in verifying results.

Solving Linear Equations

Solving linear equations involves isolating the variable of interest by performing algebraic operations such as addition, subtraction, multiplication, or division. This is a fundamental technique used to determine the exact moment or point at which a specific condition in the model is met.

Linear Function Modeling

A linear model is a mathematical tool used to represent relationships between two variables with a constant rate of change, described by the equation y = mt + b. In practical contexts, it helps to approximate real-world trends over time or other relationships where the change is steady and predictable.

Transcript

Please give Ace some feedback