- Identify variables: \(x\) for math books and \(y\) for workbooks.
- Express individual weights: math books weigh \(3.2x\), workbooks weigh \(0.8y\).
- Combine into a total weight inequality: \(3.2x + 0.8y \leq 50\).
- Final correct choice: \(\boxed{3.2x + 0.8y \leq 50}\).
Explanation
- Problem Analysis and Data Extraction
The problem asks us to determine the correct mathematical inequality representing a weight constraint for Mr. Ishimoto's order. We are given the following variables:
- \(x\): the number of new math books.
- \(y\): the number of new workbooks.
We are given the following information about weights:
- Each math book weighs \(3.2\) pounds.
- Each workbook weighs \(0.8\) pounds.
- The total weight must not exceed \(50\) pounds, which translates to an inequality where the total weight is less than or equal to \(50\).
- Setting up the Inequality
Now we construct the expression for the total weight. The weight contribution of the math books is the number of math books multiplied by the weight of each book:
\[\text{Weight}_{\text{math}} = 3.2x\]
The weight contribution of the workbooks is the number of workbooks multiplied by the weight of each book:
\[\text{Weight}_{\text{workbook}} = 0.8y\]
The total weight is the sum of these two contributions:
\[\text{Total Weight} = 3.2x + 0.8y\]
Since the total weight cannot be more than 50 pounds, we express this condition as:
\[3.2x + 0.8y \leq 50\]
- Comparing with Provided Options and Conclusion
Comparing our derived inequality, \(3.2x + 0.8y \leq 50\), with the given options:
- \(3.2x + 0.84 \leq 50\) (Incorrect: uses a constant instead of a variable \(y\))
- \(3.2x + 0.8y \leq 50\) (Matches our derived inequality)
- \(0.8x + 3.2y < 50\) (Incorrect: uses incorrect weights and strict inequality)
- \(0.8x + 3.2y \leq 50\) (Incorrect: swaps the weights of the two book types)
Therefore, the correct inequality is \(3.2x + 0.8y \leq 50\).
Examples
This type of modeling is essential for logistics and shipping, where cargo must remain within safety limits. For instance, if you are packing a suitcase for a flight with a strict weight limit, you would define weights for your items (like clothes and toiletries) as variables and create an inequality to ensure your luggage stays under the limit to avoid excess baggage fees. By mathematically modeling constraints, you can optimize how much you can pack while remaining safe and compliant.