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Calculus Optimization Question # 3 (Need Full Solution)?
Sandy is making a closed rectangular jewellery box with a square base from
two different woods. The wood for the top and bottom costs$20/m^2 . The wood
for the sides costs $30/m^2. Find the dimensions that will minimize the cost of
the wood for a volume of 4000cm^3.
answer: 12.1cm by 18.2cm by 18.2cm
V = b²h = 4000 cm³
bh = 4000/b
Cost of top and bottom = 20/10000 = .002/cm²
Cost of sides = 30/10000 = .003/cm²
Cost = C = .002(2*b²) + .003(4*bh)
C = .004b² + .012(4000/b)
C = .004b² + 48/b
dC/db = .008b - 48/b² = 0
.008b³ = 48
b³ = 6000
b = 18.171 cm
h = 4000/b² = 4000/330.193 = 12.114 cm.
Dimensions of the box are:
width and length = 18.171 cm
height = 12.114 cm
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