Katie Works Part-Time At The Fallbrook Riding Stable. She Makes $5 An Hour For Exercising Horses And S1o An Hour For Cleaning Stalls. Because Katie Is A Full-Time Student, She Cannot Work More Than 12 Hours Per Week. Graph Two Inequalities That Illustrate How Many Hours Katie Needs To Work At Each Job If She Plans To Earn No Less Than $9o Per Week. Determine
The correct answer and explanation is:
We need to model the problem with two inequalities based on the information provided:
- Katie works two jobs: exercising horses and cleaning stalls.
- For exercising horses, Katie earns $5 per hour.
- For cleaning stalls, Katie earns $10 per hour.
- Katie is a full-time student and can work no more than 12 hours per week.
- Katie wants to earn at least $90 per week.
Let’s define:
- xx = number of hours spent exercising horses.
- yy = number of hours spent cleaning stalls.
Step 1: Set up the inequalities.
Inequality 1: Total hours constraint
Katie cannot work more than 12 hours per week. Therefore, the total number of hours she works should be less than or equal to 12: x+y≤12x + y \leq 12
This inequality represents the constraint on the total number of hours worked.
Inequality 2: Earnings constraint
Katie wants to earn at least $90 per week. She earns $5 per hour for exercising horses and $10 per hour for cleaning stalls. Therefore, the total earnings can be expressed as: 5x+10y≥905x + 10y \geq 90
This inequality ensures that Katie’s earnings are no less than $90.
Step 2: Graph the inequalities.
- Graph of x+y≤12x + y \leq 12:
- This inequality is a line with slope -1 and y-intercept 12.
- The region below this line represents the set of possible values of xx and yy that satisfy the constraint on the total number of hours.
- Graph of 5x+10y≥905x + 10y \geq 90:
- This inequality simplifies to x+2y≥18x + 2y \geq 18, which is a line with slope -1/2 and y-intercept 9.
- The region above this line represents the set of possible values of xx and yy that satisfy the earnings requirement.
Step 3: Solve for feasible values of xx and yy.
The solution to this system of inequalities lies in the region where both constraints are satisfied. The points in this region represent the possible combinations of hours Katie can work at each job while earning at least $90 and not exceeding 12 hours in total.
Conclusion:
- Katie can work a variety of combinations of hours at the two jobs, as long as the total hours are ≤ 12 and her total earnings are ≥ $90. The feasible combinations of xx and yy can be found by graphing the inequalities and determining the intersection region.
- One possible combination is working 6 hours exercising horses and 6 hours cleaning stalls, which gives 5(6)+10(6)=30+60=905(6) + 10(6) = 30 + 60 = 90 dollars in earnings. There are other possibilities as well, depending on the distribution of hours between the two jobs.