Problem

Use the box model \(V~=~x(12~-~2x)(20~-~2x)\). Interpret \(\text{both}~\text{grouped}~\text{dimensions}\), state their units and positivity constraints, and give the physical domain for \(x\).

Neutral twelve-by-twenty open-box net with separated title, subtitle, top dimension 20, x-by-x corner cuts, and remaining dimensions twelve minus two x and twenty minus two x.

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What this problem is really about

Treat each parenthesized quantity as one remaining dimension of the box. Because a cut is removed from both ends of a side, each original dimension loses two copies of the cut size. The physical domain comes from requiring the height and both remaining base dimensions to stay positive, then intersecting those conditions so the tightest bound governs.

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Group 12-2x: remaining length with constraint x<6. Group 20-2x: remaining length with constraint x<10. Units: length. Uses only the looser constraint, so domain: 0<x<10.

Group 12-2x: claimed remaining length after subtracting x once. Group 20-2x: same one-end interpretation. Units: length. Constraints: x<12 and x<20. Domain: 0<x<12.

Group 12-2x: remaining 12-side length after removing x from both ends; constraint x<6. Group 20-2x: remaining 20-side length; constraint x<10. Units: length. With x>0, domain: 0<x<6.

Group 12-2x: remaining area after cuts; constraint x<6. Group 20-2x: remaining area; constraint x<10. Units: square units. With x>0, domain: 0<x<6.

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Use the box model \(V~=~x(12~-~2x)(20~-~2x)\). Interpret \(\text{both}~\text{grouped}~\text{dimensions}\), state their units and positivity constraints, and give the physical domain for \(x\).

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