Answer:
$163.54
Step-by-step explanation:
Volume of rectangular container = 10m^3
Length = 2(width)
Material for the base cost $10 per square meter
Material for the side cost $6 per square meter
Volume = L*B*H
L= 2W
V = (2W).W. H
10 = 2W^2.H
H = 10 /2W^2
H = 5/W^2
Let C(w) = cost function
C(w) = 10(L.W) + 6(2.L.H + 2.W.H)
= 10(2W.W) + 6(2.2W.H + 2.W.H)
= 10(2W^2) + 6(4W.H + 2.W.H)
= 10(2W^2) + 6(4W*5/W^2 + 2.W*5/W^2)
= 20W^2 + 6(20/W + 10/W)
= 20W^2 + 6((10+20)/W)
= 20W^2 + 6(30/W)
C(w) = 20W^2 + 180/W
To find the minimum value, differentiate C with respect to w
C'(w) = 40W - 180/W^2
Put C'(w) = 0
0 = 40W - 180/W^2
40W = 180/W^2
40W^3 = 180
W^3 = 180/40
W^3 = 4.5
W = cube rt(4.5)
W = 1.65m
C = 20(1.65)^2 + 180/1.65
C = 54.45 + 109.09
C= $163.54
Minimum cost = $163.54
