a rectangular storage container with a top is to have a volume of 36ft^3. the le

Question

a rectangular storage container with a top is to have a volume of 36ft^3. the length of the base is one and a half times the width. material for the base costs $8 per square foot and the material for the sides and top costs $4 per square foot. find the minimum cost of building the container. a rectangular storage container with a top is to have a volume of 36ft^3. the length of the base is one and a half times the width. material for the base costs $8 per square foot and the material for the sides and top costs $4 per square foot. find the minimum cost of building the container. a rectangular storage container with a top is to have a volume of 36ft^3. the length of the base is one and a half times the width. material for the base costs $8 per square foot and the material for the sides and top costs $4 per square foot. find the minimum cost of building the container.

Explanation / Answer

Let the width of the base be 'x'.

Then, the length of the base = 1.5x

Volume = length * width * height

36 = x(1.5x)h

h = 36 / 1.5x2

Material Cost, C(x) = 8x(1.5x) + 4(2) xh + 4(2)(1.5x)h + 4x(1.5x)

or C(x) = 12x2 + 192/x + 288/x + 6x2 = 18x2 + 488/x

For minimum cost, C'(x) = 0

C'(x) = 36x - 488/x2 = 0

x3 = 122 / 9

x = 2.384

C(2.384) = 18(2.384)2 + (488 / 2.384) = $ 307

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