Earlier last week I realized I needed to ship a large volume of things domestically. Of course, I decided that I wanted to do so as cheaply as possible.I first looked at USPS standard post rates. I noted that if the "combined length and girth" of a box exceeds 108", then the package becomes significantly more expensive, as it uses the "oversized" price bracket, so I resolved not to exceed this size limit. The length of a box is defined as its longest dimension, and its girth is defined as the perimeter around the rectangle to which the length is orthogonal – in other words, the girth is 2 w + 2 h.However, because I have much to ship, I also would like to maximize the volume of such a box while not exceeding this limit. This turns into a maximization problem with a constraint:Maximize V ( x ) = l w h, while satisfying 108 &gt;= l + 2 w + 2 h.This seems pretty simple, but I'm not sure how to handle the three nominally independent variables l, w, and h. Because I did not want to wait or rely on my faulty math skills, I ultimately asked Wolfram|Alpha to solve this for me, yielding a box of size 36"x18"x18". This makes intuitive sense, to a degree, but I would like a hint as to how I could have proceeded to solve this on my own.

Question

Earlier last week I realized I needed to ship a large volume of things domestically. Of course, I decided that I wanted to do so as cheaply as possible.I first looked at USPS standard post rates. I noted that if the &quot;combined length and girth&quot; of a box exceeds 108&quot;, then the package becomes significantly more expensive, as it uses the &quot;oversized&quot; price bracket, so I resolved not to exceed this size limit. The length of a box is defined as its longest dimension, and its girth is defined as the perimeter around the rectangle to which the length is orthogonal – in other words, the girth is   2  w  +  2  h.However, because I have much to ship, I also would like to maximize the volume of such a box while not exceeding this limit. This turns into a maximization problem with a constraint:Maximize   V  (  x  )  =  l  w  h, while satisfying   108  &amp;gt;=  l  +  2  w  +  2  h.This seems pretty simple, but I&#039;m not sure how to handle the three nominally independent variables   l,   w, and   h. Because I did not want to wait or rely on my faulty math skills, I ultimately asked Wolfram|Alpha to solve this for me, yielding a box of size 36&quot;x18&quot;x18&quot;. This makes intuitive sense, to a degree, but I would like a hint as to how I could have proceeded to solve this on my own.

Samantha Reid · Accepted Answer

You asked for a hint. Here are two.(1) You certainly want   l  +  2  w  +  2  h  =  108.(2) If you temporarily fix   l, you have a certain amount to make a rectangle with   w and   h. The maximum area would be to make a square out of   w and   h (i.e.,   w  =  h).

Layla Velazquez · Accepted Answer

The AM/GM inequality shows immediately that the maximum volume occurs uniquely when   l  =  2  w  =  2  h.

Earlier last week I realized I needed to ship a large volume of things domestically. Of course, I de

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