Question

Elmer had a job pulling weeds that paid $6 an hour and he made no more than $126 every week. If x is the hours he worked in a week and y is the money he made in a week, write the domain and range of the function as compound inequalities.

Domain: 0 ≤ x ≤ 21
Range: 0 ≤ y ≤ 126

Answer 1

Elmer's job paying $6 an hour results in a function with the domain of hours worked (0 to 21 hours) and the range of money made (0 to $126). These constraints form compound inequalities, which mathematically model labor-leisure budget constraints that reflect choices between work and leisure.

Step-by-step explanation:

The question is about constructing a mathematical model that describes Elmer's weekly earning through his job, which pays a certain hourly rate. Given that Elmer earns $6 an hour and cannot make more than $126 in a week, we can deduce a function that relates the hours worked (x) to the money made (y). According to this model, if Elmer works 0 hours, he would make $0, and if he works at the maximum capacity, with no more than 21 hours a week, he will make $126 (since 21 hours multiplied by $6 per hour equals $126).

The domain of this function represents the possible number of hours worked and is given by the compound inequality 0 ≤ x ≤ 21. The range represents the possible earnings and is similarly constrained by 0 ≤ y ≤ 126.

The provided scenarios illustrate the concept of a labor-leisure budget constraint, where individuals must choose between working hours and leisure time, which also includes taking care of family responsibilities. These decisions have direct implications on the income one can earn, particularly when considering government assistance programs that influence the effective wage rate through subsidies and earning reductions.