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Mathematics Forum
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| Author | Message / Information |
| lostincalc Quote | Reply | | help please.optimization problems posted on: 11/16/2006 8:31:50 PM 1. Minimizing Packaging costs-A rectengular box is to have a square base and a volume of 20 cubic feet. If the material for the base costs 30cents/sq. foot, the material for the sides costs 10 cents square foot, and the material for the top costs 20 cents 20 cents/ sq. foot, determine the dimensions of the box that can be constructed at minimum cost. 2. Minimizing costs-For its beef stew, Betty Moore Company uses aluminum containers that have the form of right circular cylinders. Find the radius and height of a container if it has a capacity of 36 cubic inches and is constructed using the least amount of metal. |
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Euler
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help please.optimization problems
replied on: 11/16/2006 9:18:14 PM Well the standard equation for volume is: V=l*w*h, but since it has a square base: V=(l^2)*h=20 Also, the surface area is: A=2(l^2)+4(l*h)=2l(l+2h), so combining them, you get: A=2l(l+(20/(1^2)))=2((l^3)+20)/(l^2) now, by taking the derivative: A'=2-80/(l^3)=0 l^3=40 --> l=2[cbrt(5)] Now, plugging back into the original V equation: h=1, so a box with length/width of 2[cbrt(5)], and height of 1, will have a volume of 20, and will cost 20*.3+20*.2+8[cbrt(5)]*.1. Hope this helps, though I am hardly awake right now, so it would be wise to check my work. |
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