The Correct Answer and Explanation is:
Of course. Here is the correct answer and a detailed explanation.
Correct Answer
a. Write and graph a system of inequalities:
Let x = the number of hours exercising horses.
Let y = the number of hours cleaning stalls.
The system of inequalities is:
- x + y ≤ 12
- 5x + 10y ≥ 60
- x ≥ 0
- y ≥ 0
The graph would show a shaded region in the first quadrant bounded by the lines x + y = 12 and 5x + 10y = 60. The solution area is the polygonal region where the individual shadings overlap.
b. Write two possible solutions:
Two possible solutions are:
- 4 hours exercising horses and 5 hours cleaning stalls.
- 2 hours exercising horses and 8 hours cleaning stalls.
Explanation
Part a: Writing and Graphing the Inequalities
To solve this problem, we first define our variables. Let ‘x’ represent the number of hours Rob spends exercising horses, and let ‘y’ represent the number of hours he spends cleaning stalls.
Next, we translate the problem’s constraints into mathematical inequalities.
- Hours Constraint: Rob can work “no more than 12 hours per week.” This means the sum of his hours exercising (x) and cleaning (y) must be less than or equal to 12. This gives us our first inequality: x + y ≤ 12.
- Earnings Constraint: He must make “at least $60 per week.” His earnings are calculated by multiplying his hours by his pay rate for each job. He earns $5 per hour for exercising horses (5x) and $10 per hour for cleaning stalls (10y). The total must be greater than or equal to $60. This gives us our second inequality: 5x + 10y ≥ 60.
- Implicit Constraints: Since the number of hours worked cannot be negative, we have two more common sense inequalities: x ≥ 0 and y ≥ 0. These confine our solution to the first quadrant of the coordinate plane.
To graph this system, you would plot the boundary lines for the two main inequalities. For x + y = 12, the intercepts are (12, 0) and (0, 12). You would draw a solid line and shade below it. For 5x + 10y = 60, the intercepts are (12, 0) and (0, 6). You would draw a solid line and shade above it. The final solution is the overlapping shaded region in the first quadrant.
Part b: Finding Two Possible Solutions
A “solution” is any ordered pair (x, y) that satisfies all the inequalities. This means we need to find a combination of hours that is 12 or less in total and earns $60 or more. Any point within the shaded solution region on the graph represents a valid solution.
- Solution 1: (4, 5) which means 4 hours exercising and 5 hours cleaning.
- Total Hours: 4 + 5 = 9. This is less than 12, so it works.
- Total Earnings: 5(4) + 10(5) = $20 + $50 = $70. This is greater than $60, so it works.
- Solution 2: (2, 8) which means 2 hours exercising and 8 hours cleaning.
- Total Hours: 2 + 8 = 10. This is less than 12, so it works.
- Total Earnings: 5(2) + 10(8) = $10 + $80 = $90. This is greater than $60, so it works.
