Problem

$C(x)~=~50~+~8\sqrt{x~-~2}$, where $x$ is hours from setup start, $C$ is cumulative cost in dollars, and production begins after a $2\text{-}\text{hour}$ setup. Interpret $x~-~2$, the threshold and domain, the outside baseline, and the scale factor.

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What this problem is really about

Interpret a contextual transformation by keeping input roles and output roles separate. The expression inside the radical resets the time clock and locates when the model begins, while the outside constant sets an output baseline in dollars. Evaluating at the threshold isolates that baseline, and the remaining multiplier describes how strongly the later growth contribution changes.

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Inside timing: x-2 is an output baseline. Threshold/domain: x=50 and x≥50. Baseline: C(50)=2 hours. Scale: 8 is an input delay.

Inside timing: x-2 is hours since production began. Threshold/domain: x=2 and x≥2. Baseline: C(2)=$50. Scale: 8 multiplies square-root growth.

Inside timing: x-2 is hours since production began. Threshold/domain: x=2 and x≥2. Baseline: C(2)=$8. Scale: 50 multiplies square-root growth.

Inside timing: x-2 is hours before production begins. Threshold/domain: x=-2 and x≥-2. Baseline: C(-2)=$50. Scale: 8 multiplies square-root growth.

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\(C(x)~=~50~+~8\sqrt{x~-~2}\), where \(x\) is hours from setup start, \(C\) is cumulative cost in dollars, and production begins after a \(2\text{-}\text{hour}\) setup. Interpret \(x~-~2\), the threshold and domain, the outside baseline, and the scale factor.

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