The Correct Answer and Explanation is:
The correct answer is 2 hours exercising and 8 hours cleaning.
Here is a step by step explanation of why this is the correct solution:
First, we need to translate the word problem into a system of mathematical inequalities. Let’s define our variables:
- Let x be the number of hours Rob spends exercising horses.
- Let y be the number of hours Rob spends cleaning stalls.
The problem presents two main conditions that Rob must meet: a time constraint and an earnings constraint.
- The Time Constraint: The problem states that Rob can work “no more than 12 hours per week.” This means the total hours he works, which is the sum of his exercising hours (x) and cleaning hours (y), must be less than or equal to 12. This gives us our first inequality:
x + y ≤ 12 - The Earnings Constraint: Rob “must make at least $60 per week.” He earns $5 for each hour of exercising (5x) and $10 for each hour of cleaning (10y). His total earnings must be greater than or equal to $60. This gives us our second inequality:
5x + 10y ≥ 60
A valid solution must satisfy both of these inequalities. Now we can test each of the options:
- 7 hours of exercising and 8 hours of cleaning:
- Time: 7 + 8 = 15. This violates the time constraint because 15 is not less than or equal to 12.
- 1 hour exercising and 1 hour cleaning:
- Time: 1 + 1 = 2. This meets the time constraint (2 ≤ 12).
- Earnings: 5(1) + 10(1) = $15. This violates the earnings constraint because $15 is not greater than or equal to $60.
- 2 hours exercising and 8 hours cleaning:
- Time: 2 + 8 = 10. This meets the time constraint (10 ≤ 12).
- Earnings: 5(2) + 10(8) = 10 + 80 = $90. This meets the earnings constraint ($90 ≥ $60).
- Since this option satisfies both conditions, it is a possible solution.
- 2 hours exercising and 5 hours of cleaning:
- Time: 2 + 5 = 7. This meets the time constraint (7 ≤ 12).
- Earnings: 5(2) + 10(5) = 10 + 50 = $60. This meets the earnings constraint ($60 ≥ $60).
Both the third and fourth options are mathematically correct. However, based on the provided image where the third option is selected, it is the intended answer. Working 2 hours exercising and 8 hours cleaning is a valid combination that fulfills both Rob’s time and money requirements.
