Cost = cost of top + cost of bottom + cost of 4 sides
= 2L^2 + 2L^2 + 3(4Lh)
= 4L^2 + 12Lh
but L^2 h = 20
h = 20/L^2
so Cost = 4L^2 + 12L(20/L^2)
= 4L^2 + 240/L
d(Cost)/dL = 8L - 240/L^2 = 0 for max/min of Cost
8L = 240/L^2
L^3 = 30
L = 3.107
then h = 20/3.107^2 = 2.07
L^2*h=20 Divide with (L^2)
Area of square is L^2
Areao of rectacangle L*h
Total price=2*Area of square*(2$)+4*Area of rectacangle*(3$)
Price have minimum where is (dP/dL)=0
8*L=240/(L^2) Divide with 8
L=30/(L^2) Multiply with L^2
L=third root of 30
=20/(third root of 30)^2
h=20/third root of 900
Proof that is minimum
Function have minimum when:
(dP/dL)=0 and (d^2P)/dL^2>0
First derivation=0 and
second derivation higher of zero
(d^2P)/dL^2=Derivation of first derivation
8+480/L^3 is always higher of zero.
Function have minimum.
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