- Identify the variable \(w\) as the amount of water in the barrel.
- Interpret "maximum capacity of 55 gallons" as "at most 55 gallons," which translates to the inequality \(\leq\).
- Conclude that the condition is represented by \(w \leq 55\).
- The final answer is \(\boxed{w \leq 55}\).
Explanation
- Analysis of the Problem
The problem asks us to translate a real-world constraint into a mathematical inequality. We are given that Josie has a barrel with a maximum capacity of 55 gallons of water. Let \(w\) represent the amount of water in the barrel in gallons. The term "maximum" indicates the highest possible value the variable \(w\) can take. Since the barrel can hold at most 55 gallons, the amount of water \(w\) can be any value from 0 up to 55, inclusive.
- Translating the Constraint to an Inequality
In mathematics, phrases like "maximum," "at most," or "no more than" are translated into the "less than or equal to" symbol, denoted by \(\leq\). This symbol implies that the quantity is either smaller than the limit or exactly equal to the limit. Therefore, the statement "a maximum of 55 gallons" means that the amount of water \(w\) must be less than or equal to 55. We represent this mathematically as \(w \leq 55\).
- Conclusion
We have the options:
- \(w > 55\)
- \(w < 55\)
- \(w \leq 55\)
- \(w \geq 55\)
The inequality \(w \leq 55\) matches our derived condition exactly. It represents the constraint that the volume of water \(w\) cannot exceed the capacity of the barrel, 55 gallons. Thus, the correct inequality is \(w \leq 55\).
Examples
In daily life, capacity constraints appear in many forms. For example, an elevator might have a "maximum capacity of 1000 lbs". If \(p\) is the total weight of passengers in the elevator, the safety limit is represented by \(p \leq 1000\). Understanding these inequalities helps in designing systems and ensuring safety, whether it is for water storage, traffic limits on a bridge, or weight limits for a shipping container.