Optimization: Best Packaging Method for Juice Box

[wind]

Hi, this question is confusing me, can someone help?

A juice manufactuer is studing the most economical shape to use for a beverage container. Each unit will contain 355cm³ of juice. The manufacturer is considering a cylinder versus a rectangular prism with a comfortable hand-hels depth of 4cm. Which method of packaging, the can or juice box, will use the minimum amount of materals?

Cylinder

355=pir²4
355/4pi=r
5.316=r

SA= 2pirh+2pir²
there are no unknowns

so how does optimization come in to this?

Thanks

 

[skeeter]

I do not think that calculus is required to solve this problem.

making some assumptions here because of the limited given information (that's always dangerous) ...

the 4 cm "hand-held" depth leads me to believe the rectangular prism will have dimensions of 4 x 4 x h = 355, so h = 355/16 and the total surface area of the rectangular container is 16h + 32 = 355 + 32 = 387 cm².

the 4 cm "hand-held" depth for the cylinder would mean that the cylinder's diameter is 4 cm ... so r = 2 cm.

pi*2²*h = 355, so h = 355/(4pi)
surface area of the cylindrical container is 2(pi*2²) + 2pi*2*355/(4pi) = 8pi + 355 = approx 380 cm²

looks like the cylindrical container wins (as it did when soda/beer cans were initially designed ... I believe 355 cm³ is the industry standard 12 fluid ounces of volume).