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3 problems match “critical points”. Clear

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1. Corner squares for maximum box volume ●●○○○ 20 ptsAn open-top box is made from a square sheet of metal with sides of 20 cm by cutting equal squares from the corners and folding up the sides. Find the side length (in cm) of the square cut from each corner so that the volume of the box is maximised. Round to two decimal places.
2. Cuboid of minimal surface area ●●●○○ 50 ptsA juice box is a rectangular box (cuboid) with volume $200\ \text{cm}^3$ ($200$ ml). Find the length $l$, width $w$ and height $h$ (in cm) of the box with the least possible surface area. Round each answer to five decimal places.
3. Stiffest rectangular beam from a log ●●●○○ 50 ptsThe stiffness of a beam is proportional to its width times the cube of its depth. Find the width and depth of the stiffest rectangular beam that can be cut from a cylindrical log of diameter 12 inches. Give the width and depth in inches. Round to three decimal places.