Concept explainers
Life Cycle Hypothesis An individual’s income varies with his or her age. The following table shows the median income I of males of different age groups within the United States for 2012. For each age group, let the class midpoint represent the independent variable, . For the class “65 years and older,” we will assume that the class midpoint is .
(a) Use a graphing utility to draw a
(b) Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income.
(c) Use the function found in part (b) to determine the age at which an individual can expect to earn the most income.
(d) Use the function found in part (b) to predict the peak income earned.
(e) With a graphing utility, graph the quadratic function of best fit on the scatter diagram.
To calculate:
a. Graph a scattered diagram and determine the type of relation that exists between the 2 variables.
b. Find the quadratic function of best fit.
c. Determine the age at which the individual gets maximum income.
d. Find the maximum income earned.
e. Graph the quadratic function of best fit.
Answer to Problem 25AYU
a. The graph is given below.
b. The equation of best fit is
c. People can expect a maximum income at around the age of 48.
d. The maximum income earned is about .
e. The graph is given below.
Explanation of Solution
Given:
The median income, of males of different age groups is given below. The class midpoint of the age group represents the variables.
Formula used:
For a quadratic equation , if is positive, then the vertex is the minimum point and if is negative, the vertex is the maximum point.
Calculation:
We can draw the scatter diagram using Microsoft Excel.
Thus, on entering the and the values on excel, we have to choose the Scatter diagram form the insert option.
Then for getting the equation of best fit, we have to choose the Layout option and then Trendline and then more Trendline option. Then choose the option polynomial and then display the equation.
Thus, we get the scatter diagram with the equation of best fit as
a. The scatter diagram is drawn above and we can see that the given variables exhibit a non-linear polynomial relationship.
b. The equation of best fit is
c. We can see that the equation of best fit is a quadratic equation with
Here, is negative, therefore, the vertex is the maximum point.
The value for maximum is
Thus, at around the age of 48, people can expect a maximum income.
d. The maximum income earned is
The maximum income earned is about .
e. The graph is drawn above.
Chapter 3 Solutions
Precalculus Enhanced with Graphing Utilities
Additional Math Textbook Solutions
Thomas' Calculus: Early Transcendentals (14th Edition)
Precalculus (10th Edition)
Calculus & Its Applications (14th Edition)
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
Precalculus
Calculus: Early Transcendentals (2nd Edition)
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning