determine the dimensions of the box that can be constructed at minimum cost.

label Calculus
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Jul 17th, 2015

Thank you for the opportunity to help you with your question!

let the dimensions of base = x ft

let the height be h cm

volume, V = x^2 h = 20

h = 20 / x^2

area of base = x^2

area of top = x^2

area of 4 sides = 4xh

Total Cost, C = 0.3x^2 + 0.2 x^2 + 0.1(4xh)

C = 0.5x^2 + 0.4 xh

substitute h = 20 /x^2

C = 0.5 x^2 + 8 / x

differentiating

dC/dx = x - 8 x^2

inorder to minimize cost , equate dC/dx to 0

x - 8/x^2 = 0

x^3 = 8

x = 2 ft

h = 20 /4 = 5

so dimensions of box are 2 x 2 x 5 ft

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jul 17th, 2015

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