rob works part time at the fallbrook riding stable. He makes $5 an hour exercising horses and $10 hour cleaning stalls. Because Rob is a full time student he can work no more than 12 hours per week. however he must make $60 per week. write a system of inequalitites
The Correct Answer and Explanation is :
To model Rob’s situation with a system of inequalities, we will let:
- ( x ) represent the number of hours Rob spends exercising horses.
- ( y ) represent the number of hours he spends cleaning stalls.
Given the information provided, we can formulate the following inequalities:
- Total Hours Constraint: Rob can work no more than 12 hours per week.
[
x + y \leq 12
] - Earnings Requirement: Rob must earn at least $60 per week. Since he earns $5 for each hour spent exercising horses and $10 for each hour cleaning stalls, we can express his earnings as:
[
5x + 10y \geq 60
] - Non-negativity Constraints: Rob cannot work negative hours.
[
x \geq 0
]
[
y \geq 0
]
Putting it all together, we have the following system of inequalities:
[
\begin{align*}
- & \quad x + y \leq 12 \
- & \quad 5x + 10y \geq 60 \
- & \quad x \geq 0 \
- & \quad y \geq 0 \
\end{align*}
]
Explanation
Rob’s work scenario is constrained by both the number of hours he can work and the amount of money he needs to make. The first inequality ( x + y \leq 12 ) reflects the limit of 12 hours per week, ensuring that he balances his work and study commitments effectively. This inequality means that the combined hours of exercising and cleaning must not exceed 12, which helps him manage his time efficiently as a full-time student.
The second inequality ( 5x + 10y \geq 60 ) ensures that Rob’s total earnings meet or exceed the required $60. Since cleaning stalls pays more, Rob might prioritize those hours or balance them strategically with exercising to maximize his earnings within the time constraints.
The non-negativity constraints ( x \geq 0 ) and ( y \geq 0 ) ensure that the hours worked cannot be negative, reinforcing the realistic nature of the problem.
By analyzing these inequalities, Rob can determine feasible combinations of hours spent on both tasks, helping him achieve his financial goals while accommodating his educational responsibilities.