Boxes of maximum volume

Solve the first problem 18 problems

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.
Minimum surface area of a square prism ●●○○○ 20 ptsA closed rectangular prism has a square base with side length $L$ and height $H$ . Its volume is $10000$ . Find the minimum possible surface area. Round to three decimal places.
Open-top box with minimum surface area ●●○○○ 20 ptsA container company makes an open-top rectangular box with a square base. The box must have a volume of $108\ \text{cm}^3$ . Let the side length of the square base be $x$ cm and the height be $h$ cm. (1) Find the value of $x$ that gives the minimum outer surface area. (2) Find the minimum surface area, in $\text{cm}^2$ .
Largest edge of a box ●●●○○ 50 ptsAn $m\times n\times p$ rectangular box has half the volume of an $(m + 2)\times(n + 2)\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are positive integers with $m\le n\le p$ . Find the largest possible value of $p$ .
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.
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.

Boxes of maximum volume

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