Question

Cuboid containers (open at the top) should be examined with regard to their volume. The figure below shows a network of such containers (x ∈ Df). Determine a function ƒ (assignment rule and definition area D) that describes the volume of these containers and calculate the volume of such a container if the content of the base area is 16 dm². Show that this function f has neither a local maximum nor a global maximum

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Hank

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55 Answers

A cuboid container open at the top can be considered as a box with length l, width w, and height h.
The volume V of such a container can be given by the function V = lwh.
If the content of the base area is given (let’s denote it by A), and assuming the base is a square (so l = w), we can express l and w in terms of A as l = w = sqrt(A).
The volume function then simplifies to V = Ah, where h is the height of the container.
Given that A = 16 dm², the volume of the container for a specific height h would be V = 16h.
The derivative of V(h) = 16h with respect to h is dV/dh = 16, which is a constant.
This means that the function V(h) is a linear function with a constant slope, and it increases as h increases. Therefore, it doesn’t have a maximum value (neither local nor global), as it will continue to increase as long as h increases.

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