How wide is the chasm between what men and women earn in the workplace? According to a 2015 analysis from a national women's group, women lose $435,049 over the course of a career because of the pay gap. The bar graph to the right shows the average earnings in the United States for men and women at ages 23 and 55. This exercise involves the graphs of models for the data shown in the rectangular coordinate system to the right. Complete parts (a) through (c) below. A side-by-side bar graph labeled "Average Yearly Earnings in the U.S., by Gender and Age" has a horizontal axis labeled "Age" with the classes 23 and 55 and a vertical axis labeled "Average Yearly Earnings in thousands of dollars" from 0 to 70 in increments of 5. Two vertical bars are above each of the horizontal axis labels, where the left bar represents men and the right bar represents women. The heights of the bars are as follows, where the age is listed first, and the heights of the bars are listed next from left to right: 23, 29 and 24; 55, 61 and 35. Average Yearly Earnings in the U.S., by Gender and Age 23 Age 55 0 10 20 30 40 50 60 70 Average Yearly Earnings ($1000s) Men Women 0 10 20 30 40 50 $0 $10 $20 $30 $40 $50 $60 $70 Years after Age 23 Average Yearly Earnings ($1000s) left parenthesis 0 comma 24 right parenthesisleft parenthesis 30 comma 35 right parenthesis (0,29) (30,61) Women Men
A coordinate system has a horizontal x-axis labeled Years after age 23 from 0 to 50 in increments of 10 and a vertical y-axis labeled Average Yearly Earnings in thousands of dollars from 0 to 70 in increments of 10. A line labeled men rises from left to right and passes through the labeled points (0, 29) and (30, 61). A line labeled women rises from left to right and passes through the labeled points (0, 24) and (30, 35). Question content area bottom Part 1 a. Use the two points for men shown on the graph to find a function in the form Upper M left parenthesis x right parenthesis equals mx plus b that models average yearly earnings for men x years after age 23. Upper M left parenthesis x right parenthesis equals enter your response here (Use integers or decimals for any numbers in the expression. Round to two decimal places as needed.) Part 2 b. Use the two points for women shown on the graph to find a function in the form Upper W left parenthesis x right parenthesis equals mx plus b that models average yearly earnings for women x years after age 23. Upper W left parenthesis x right parenthesis equals enter your response here (Use integers or decimals for any numbers in the expression. Round to two decimal places as needed.) Part 3 c. Use the models in parts (a) and (b) to find the average yearly earnings for men and women at age 32. How are these values shown on the graphs of the models for the data? What is the difference between earnings for men and women at that age? Start by finding the average yearly earnings for men at age 32. Upper M left parenthesis age equals 32 right parenthesis equals$ enter your response here thousand (Type an integer or decimal rounded to one decimal place as needed.) Part 4 Next find the average yearly earnings for women at age 32. Upper W left parenthesis age equals 32 right parenthesis equals$ enter your response here thousand (Type an integer or decimal rounded to one decimal place as needed.) Part 5 How are these values shown on the graphs of the models for the data? A. By the points (23,61) and (23,35) respectively. B. By the points (9,38.6) and (9,27.3) respectively. C. By the points (32,38.6) and (32,27.3) respectively. D. By the points (0,61) and (0,35) respectively. Part 6 The difference in yearly earnings at age 32 is $ enter your response here thousand. (Type an integer or decimal rounded to one decimal place as needed.) Upper M left parenthesis x right parenthesis equals(Use integers or decimals for any numbers in the expression. Round to two decimal places as needed.)Upper W left parenthesis x right parenthesis equals(Use integers or decimals for any numbers in the expression. Round to two decimal places as needed.)Upper M left parenthesis age equals 32 right parenthesis equals$(Type an integer or decimal rounded to one decimal place as needed.)Upper W left parenthesis age equals 32 right parenthesis equals$(Type an integer or decimal rounded to one decimal place as needed.)How are these values shown on the graphs of the models for the data?The difference in yearly earnings at age 32 is $(Type an integer or decimal rounded to one decimal place as needed.)
To solve the problem, we will follow the steps outlined in the questions.
We have two points for men:
We can use the slope-intercept form of a linear equation, ( M(x) = mx + b ).
Calculate the slope (m): [ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{61 - 29}{30 - 0} = \frac{32}{30} = \frac{16}{15} \approx 1.067 ]
Use one of the points to find b: Using the point (0, 29): [ M(0) = m(0) + b \implies 29 = 0 + b \implies b = 29 ]
Write the function: [ M(x) = \frac{16}{15}x + 29 ]
We have two points for women:
Calculate the slope (m): [ m = \frac{35 - 24}{30 - 0} = \frac{11}{30} \approx 0.367 ]
Use one of the points to find b: Using the point (0, 24): [ W(0) = m(0) + b \implies 24 = 0 + b \implies b = 24 ]
Write the function: [ W(x) = \frac{11}{30}x + 24 ]
To find ( M(32 - 23) = M(9) ): [ M(9) = \frac{16}{15}(9) + 29 = \frac{144}{15} + 29 = 9.6 + 29 = 38.6 ] So, ( M(32) = 38.6 ) thousand dollars.
To find ( W(32 - 23) = W(9) ): [ W(9) = \frac{11}{30}(9) + 24 = \frac{99}{30} + 24 = 3.3 + 24 = 27.3 ] So, ( W(32) = 27.3 ) thousand dollars.
The correct answer is: C. By the points (9, 38.6) and (9, 27.3) respectively.
To find the difference: [ \text{Difference} = M(32) - W(32) = 38.6 - 27.3 = 11.3 ] So, the difference in yearly earnings at age 32 is $11.3 thousand.
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