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# Calculus minimal cost

### Customer Question

A rectangular box is to have a square base and V=160ft cubed. If material for base costs 20 cents a square foot, material for sides costs 10 cents per square foot and material for top costs 30 cents per square foot, determine the dimensions of the box that can be made at the minimal cost
Submitted: 11 years ago.
Category: Homework
Expert:  skywalker replied 11 years ago.
V=x^2*h=160 x is the base side
then total cost for base will be *x^2*20 ( two for upward it is 30*x^2
for the sides ( 4 sides) 4*10*h

4*30*h
TC= 20x^2+30x^2+ 40h
where 160=h*x^2
then
TC= 50x^2+ 6400/x^2
if we take differntiano in order to minimze the cost and equate to 0
0=100x-6400/x^3
then
100x=6400/x^3
x^4=640
x=5.02
then h=6.32

x=
Customer: replied 11 years ago.
Reply to skywalker's Post: so what are the dimensions of the box then?
Expert:  skywalker replied 11 years ago.
the dimension are x=5.02 ft for one side of the base
and the height is 6.32
dimensions are (x,h)=(5.02,6.32)
Expert:  jacket replied 11 years ago.

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Expert:  xarqi replied 11 years ago.
Jacket's answer is the correct one.

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