SOLUTION: A rectangular box with square base and a fixed volume (V) is to be made from two types of material. The material for the top and bottom costs t cents per square inch and the materi

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Question 949300: A rectangular box with square base and a fixed volume (V) is to be made from two types of material. The material for the top and bottom costs t cents per square inch and the material for the sides costs s cents per square inch. Let the length of the base be x and the height be h. Find x and h(in terms of V, t, s) so that the cost is minimal.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
V=x%2Ax%2Ah=x%5E2%2Ah
C=2x%5E2%2At%2B4xh%2As
So then,
h=V%2Fx%5E2
Substituting,
C=2x%5E2%2At%2B4x%28V%2Fx%5E2%29s
C=2tx%5E2%2B%284Vs%29%2Fx
To minimize C, take the derivative and set it equal to zero.
dC%2Fdx=4tx-%284Vs%29%2Fx%5E2=0
%284Vs%29%2Fx%5E2=4tx
x%5E3=%28Vs%29%2Ft
highlight%28x=%28%28Vs%29%2Ft%29%5E%281%2F3%29%29
and
h=V%2Fx%5E2
x%5E2=%28%28Vs%29%2Ft%29%5E%282%2F3%29=V%5E%282%2F3%29%2A%28s%2Ft%29%5E%282%2F3%29
1%2Fx%5E2=V%5E%28-2%2F3%29%2A%28t%2Fs%29%5E%282%2F3%29
highlight%28h=V%5E%281%2F3%29%2A%28t%2Fs%29%5E%282%2F3%29%29