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Difficulty: HardLinear Inequalities in One Variable

A shipping container has a maximum weight capacity of 24,15024,150 kilograms. The container is loaded with 1212 machinery units, each weighing 1,1501,150 kilograms. The remaining space will be filled with packing crates, each weighing 180180 kilograms. A safety regulation requires that a clearance weight of at least 15%15\% of the total loaded weight (the combined weight of the machinery units and the packing crates) must be left unused. What is the maximum number of packing crates that can be loaded into the container?

Answer: 40

Answer

The maximum number of packing crates that can be loaded is 40.
By setting up the inequality representing the physical constraints, we find that the number of packing crates, xx, must satisfy x≤40x \le 40. Since the question asks for the maximum number of packing crates, the maximum value is 40.

Step-by-Step Solution

1
Calculate the constant weight of the machinery units.
The total weight of the 1212 machinery units is 12×1,150=13,80012 \times 1,150 = 13,800 kilograms.
This establishes the base weight that is already loaded in the container.
2
Define the variable and write the expression for the total loaded weight.
Let xx be the number of packing crates. The total loaded weight is 13,800+180x13,800 + 180x kilograms.
This represents the combined weight of the machinery and the crates in terms of the variable xx.
3
Set up the inequality representing the safety clearance requirement.
24,150−(13,800+180x)≥0.15(13,800+180x)24,150 - (13,800 + 180x) \ge 0.15(13,800 + 180x)
The unused weight capacity (maximum capacity minus loaded weight) must be at least 15%15\% of the loaded weight.
4
Solve the inequality for xx.
24,150≥1.15(13,800+180x)  ⟹  21,000≥13,800+180x  ⟹  7,200≥180x  ⟹  x≤4024,150 \ge 1.15(13,800 + 180x) \implies 21,000 \ge 13,800 + 180x \implies 7,200 \ge 180x \implies x \le 40.
Isolating xx gives the range of allowable values for the number of packing crates.

Key Concept

Formulating and solving linear inequalities in one variable based on real-world constraints.
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Topics

Linear Inequalities in One Variable

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