Problem

$C(x)~=~(500~+~8x)/x$ models average cost in dollars per item when $x$ is a positive whole number of items. Rewrite the model to separate its terms, interpret each term and the domain, and describe the limiting cost per item.

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What this problem is really about

Separate the numerator term by term before interpreting the average. That rewrite distinguishes the fixed cost spread across all items from the per-item contribution, and it also makes the long-run behavior immediate: the fixed share shrinks as production grows. Keep the real-world domain to positive whole numbers of items.

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C(x)=500/x+8. The term 500/x is the fixed-cost share per item, 8 is the variable cost per item, x is a positive whole number, and C approaches $8 per item as x increases.

C(x)=508/x. The term 508/x is the total-cost share per item, there is no variable cost, x is a positive whole number, and C approaches $0 per item as x increases.

C(x)=500/x+8. The term 500/x is the variable cost per item, 8 is the fixed total cost, x is any nonzero real number, and C approaches $8 total as x increases.

C(x)=500+8/x. The term 8/x is the fixed-cost share per item, 500 is the variable cost per item, x is a positive whole number, and C approaches $500 per item as x increases.

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\(C(x)~=~(500~+~8x)/x\) models average cost in dollars per item when \(x\) is a positive whole number of items. Rewrite the model to separate its terms, interpret each term and the domain, and describe the limiting cost per item.

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