SOLUTION: A metal box with a square bottom and top is to contain 768 cubic centimeters. The bottom material must be stronger than the rest of the box and costs four cents per square centimet

Question 1155324: A metal box with a square bottom and top is to contain 768 cubic centimeters. The bottom material must be stronger than the rest of the box and costs four cents per square centimeter. Material for the sides and tops is less expensive and costs two cents per square centimeter. Find the dimensions of the box that satisfy the volume requirements at a minimum cost of materials. What is this minimum cost?
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A metal box with a square bottom and top is to contain 768 cubic centimeters.
The bottom material must be stronger than the rest of the box and costs four cents per square centimeter.
Material for the sides and tops is less expensive and costs two cents per square centimeter.
Find the dimensions of the box that satisfy the volume requirements at a minimum cost of materials. What is this minimum cost?
:
let s = size of the side of the square bottom and square top
let h = the height of the box
:
s^2 * h = the volume
s^2 * h = 768
h =
:
Surface area = bottom area + 4 sides area + top area
SA = s^2 + 4(hs) + s^2
replace h
SA = s^2 + 4(*s) + s^2
cancel s
SA = s^2 + 4() + s^2
SA = s^2 + () + s^2
find the cost
cost = bottom + 4 sides + top
Cost = .04(s^2) + .02() + .02s^2
Cost = (.04s^2) + () + .02s^2
Cost = (.06s^2) + ()
graph this equation

Minimum occurs when s = 8 cm
Find h
h =
h = 12 cm
dimensions then: 8 by 8 by 12 cm
:
Find the actual cost
Cost = .06(8^2) + .02(4(8*12))
Cost = 3.84 + 7.68
cost ~ $11.52 is the min cost of the box (green line)

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