l + w + h <= 56
h = w
l + 2h <= 56
For largest volume, l + 2h = 56 --> L = 56 - 2h
V = lwh
V = (56 - 2h)(h)(h)
V = -2h^3 + 56h^2
dV/dh = -6h^2 + 112h = 0
-6h^2 + 112h = 0
h = 112/6
So, h = 56/3 inches
Therefore, w = 56/3 inches
l = 56 - 2h
l = 56 - 2(56/3)
l = 56 - 112/3
l = 56/3 inches
So, the dimensions are :
Length = 56/3 inches
Width = 56/3 inches
Height = 56/3 inches
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C(x) = 16900 + 100x + 0.01x^2
So, average cost = Ca(x) = (16900 + 100x + 0.01x^2) / x
Ca(x) = 16900/x + 100 + 0.01x
dCa/dx = -16900/x^2 + 0.01 = 0
16900/x^2 = 0.01
x^2 = 16900/0.01
x^2 = 1690000
x = 1300 ----> ANSWER
Average cost = 16900/1300 + 100 + 0.01(1300)
Resulting average cost = $126 ---> ANSWER
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3)
L + 2W + 2H <= 126
L + 2W + 2H = 126
If front face is square, then W = H
L + 4W = 126
L = 126 - 4W
Volume V = LWH
V = (126 - 4w)(w)(w)
V = -4w^3 + 126w^2
dV/dw = -12w^2 + 252w = 0
w = 252/12 ---> w = 21
So, h = 21
L = 126 - 4w --> 126 - 4(21)
L = 42
So, max volume is : LWH --> 42*21*21
V(max) = 18522 in^3 -----> ANSWER
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