Minimisation question - volume

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In summary, the question is asking how to choose the height and depth of a cuboid-shaped coal box that will minimize the visible surface area while still being able to contain a certain volume of coal. This problem can be approached using either the method of reducing variables or Lagrange multipliers. Both methods involve solving a system of equations to find the optimal values for the height, depth, and width of the box.
  • #1
nathangrand
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A coal box, in the shape of a cuboid, is to be placed flush against a wall so that only its top, front and two ends are visible. How should the height h and the depth d be chosen so a to minimise the visible surface area A under the constraint that the box must be able to contain atleast a certain volume V of coal?

Here's how far I've got:

V=hwd (where w is the width)
Visible surafce area = 2hd + homework +wd
 
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  • #2
There are two ways to approach this. One is to use hwd= V to reduce the number of variables: h= V/wd so the function to be minimized becomes 2hd + homework +wd = 2(V/wd)d+ (V/wd)w+ wd= 2V/w+ V/d+ wd. Now take the partial derivatives with respect to w and d and set them equal to 0.

The other way is to use Lagrange multipliers. The gradient of the target function is <2h+ w, 2d+ w, h+ d> where the components are the derivatives with respect to d, h, and w in that order. The gradient of the constraint function, with derivatives in the same order, is <hw, wd, hd>. At the optimal point, we must have [itex]<2h+ w, 2d+ w, h+ d>= \lambda<hw, wd, hd>[/itex]. That is, we must have [itex]2h+ w= \lambda hw[/itex], [itex]2d+w= \lambda wd[/itex], and [itex]h+ d= \lambda hd[/itex], which, together with hwd= V, give four equations for d, h, w, and [itex]\lambda[/itex].

Since a value of [itex]\lambda[/itex] is not necessary for the solution, I find it is often best to eliminate [itex]\lambda[/itex] by dividing one equation by another.
 
  • #3
Thanks - I'll give both methods a go and see how I get on!
 

1. What is a minimisation question in terms of volume?

A minimisation question in terms of volume is a scientific inquiry that aims to determine the smallest possible volume of a given object or substance under certain conditions.

2. How is volume minimisation important in scientific research?

Volume minimisation is important in scientific research because it allows scientists to understand the fundamental properties of a substance or object by studying its smallest possible unit. This can provide insights into its structure, behavior, and potential applications.

3. What are some examples of volume minimisation in scientific studies?

Some examples of volume minimisation in scientific studies include determining the smallest possible volume of a cell, molecule, or particle; investigating the minimum volume required for a chemical reaction to occur; and studying the impact of volume reduction on the properties of materials.

4. What methods are used to minimise volume in scientific experiments?

There are various methods used to minimise volume in scientific experiments, including compression, freezing, and manipulation of environmental conditions. Other techniques such as microscale and nanoscale fabrication can also be used to create smaller volumes for study.

5. What are the potential applications of volume minimisation in different fields of science?

Volume minimisation has a wide range of potential applications in fields such as materials science, chemistry, biology, and engineering. For example, it can be used to develop more efficient and compact devices, create new materials with unique properties, and understand the inner workings of biological systems at a molecular level.

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