Why Is There Only One Stationary Point for the Volume of This Box?

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Homework Help Overview

The problem involves determining the stationary points of the volume function for a box defined by V=4x^3 -66x^2 +216x. The inquiry focuses on why there is only one value of x that results in a stationary point for this volume function.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the derivative of the volume function and the resulting quadratic equation to find stationary points. There is a consideration of the physical implications of the volume values at those points, particularly questioning the feasibility of negative volume.

Discussion Status

Participants are exploring the implications of the calculated stationary points and discussing the physical constraints of the problem. There is acknowledgment of the reasoning that leads to identifying x=2 as the only viable stationary point due to the context of the problem.

Contextual Notes

Participants note that the physical context of the problem imposes constraints on the values of x, specifically regarding the requirement for volume to be non-negative.

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Homework Statement


The volume of a box is given by V=4x^3 -66x^2 +216x. Explain why there is only one value of x for which V is stationary?


Homework Equations





The Attempt at a Solution


dV/dx = 12x^2 -132x +216 at a stationary point x^2-11x+18=0 so x=2 or 9
 
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You need to check if those solutions are reasonable since you're talking about a physical situation. What are the volumes for those two points?
 
I got a volume of -486 for x=9 and 200 for x=2. I think that the one stationary point must be at x=2 because a box can't have a negative volume. Is this correct?
 
Yes, that's exactly it.
 
Thanks for the help
 

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