AMC 12 Mock Exam 2 Problem 16

Problem 16 of 25

A rectangular box with an open top (no lid) is to have a volume of exactly $32$ cubic feet. Its bottom and four sides are all made from the same sheet material. What is the least possible total area, in square feet, of material needed to make the box?

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Solution

Let the base be $x\times y$ and the height $h$ , so $xyh=32$ . The area of material is $S=xy+2xh+2yh$ . By AM–GM, $S\ge3\sqrt[3]{xy\cdot2xh\cdot2yh}=3\sqrt[3]{4(xyh)^2}=3\sqrt[3]{4\cdot1024}=3\cdot16=48.$ Equality holds when $xy=2xh=2yh$ , i.e., $x=y=2h$ ; then $4h^3=32$ , so $h=2$ and $x=y=4$ , which gives $S=16+16+16=48$ . The minimum is $\boxed{48}$ .

Tagged: optimization by AM-GM (open box)

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