The cost of materials for the least expensive such container is 245.31 dollars .
Suppose the width of container is x (m),
length of the base of container is 2x (m)
the base area of container is (length × width)sq. m i.e 2x² meter².
Since the volume of container is 10 m³.
the height of container = area of container/ volume of container
height of container= 10/2x² m = 5/x² m
The cost of making such container is cost of base = 2x²*10 = 20x².
cost of sides of container = (2×2x × 5/x² + 2× x × 5/x²)×6 = 180/x
The overall cost is hence the sum of the base and the sides: f(x) = 20x² + 180/x
The get the minimum,
df(x)/dx = 20×(2x - 9/x²) = 0
so 2x = 9/x^2 => 2x = 2.0800 ==> x = 1.651 m
f(x) = 245.31 dollars
To learn more about Maxima or minima, refer:
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