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Suppliers sell centimetre cubes to schools in packages shaped like square based prisms. Determine the dimensions of the package that would require the least material to hold 1200 cubes?

Jun 8, 2021

#1
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*** edit ****

We need dimensions that are in whole number integers to fit the cubes in the box

10 x 10 x 12    would work      surface area   = 680 cm2

Jun 8, 2021
edited by ElectricPavlov  Jun 8, 2021
edited by ElectricPavlov  Jun 8, 2021
#2
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Let the dimensions of the package be $$a$$ by $$b$$ by $$c$$.  Thus, the volume is $$abc$$

The total material required to create the package would be the surface area, namely, $$2(ab+bc+ac)$$.  With this, we seek to find an inequality that links $$ab+bc+ac$$ to $$abc$$, and the AM-GM Inequality does just that.

By the AM-GM Inequality for 3 variables, $$\frac{ab+bc+ac}{3}≥\sqrt[3]{abc^2}$$, so $$\frac{ab+bc+ac}{3}≥\sqrt[3]{1440000}$$.

Simplifying, we have $$ab+bc+ac≥60\sqrt[3]{180}$$.  The minimum namely, the right side of the above equation is achieved when the equality condition of the AM-GM is met, or when ab=bc=ac.  This simplifies to a=b=c, or a=b=c= $$2\sqrt[3]{150}$$, from abc = 1200

My answer for the dimensions would be $$2\sqrt[3]{150}$$ by $$2\sqrt[3]{150}$$ by $$2\sqrt[3]{150}$$.

Jun 8, 2021