SOLUTION: A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs $0.35/ft2, the material for the sides costs $0.05/ft2, and the material for th

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Question 1028760: A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs $0.35/ft2, the material for the sides costs $0.05/ft2, and the material for the top costs $0.15/ft2, determine the dimensions of the box that can be constructed at minimum cost.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A rectangular box is to have a square base and a volume of 40 ft3.
If the material for the base costs $0.35/ft2, the material for the sides costs $0.05/ft2, and the material for the top costs $0.15/ft2, determine the dimensions of the box that can be constructed at minimum cost.
:
let x = the length of the side of the square base
then
40%2Fx%5E2 = the height of the box
:
Area of the top and bottom will be x^2
Area of each of the 4 sides will be :
x%2840%2Fx%5E2%29 simplifies to 40%2Fx
Fours sides area: 4*40%2Fx = 160%2Fx
:
The cost equation
Cost = base cost + side cost + top cost
C(x) = .35x%5E2+%2B+.05%28160%2Fx%29+%2B+.15x%5E2
C(x) = .50x%5E2+%2B+8%2Fx
Graph this equation to find the min cost
+graph%28+300%2C+200%2C+-2%2C+5%2C+-4%2C+20%2C+.5x%5E2%2B%288%2Fx%29%29+
minimum cots occurs when x = 1.7 ft, the length and width of the box
Find the height
h = 40%2F1.7%5E2
h = 13.8 ft
:
The box dimension for min cost: 1.7 by 1.7 by 13.8