Calculus (minima) problem

steveb

Joined Jul 3, 2008
2,431
If you look at the spreadsheet you can see clearly that for a, b =18, 24 the minimum does not occur at r=0

In fact since only the walkway contributes to the variation in area for values of r less than 3 we want to maximise the side with the area b*r and miniimise the side with the area 3*a
Similarly for r>3 we want to minimise rb and maximise 3a.

This also shows up clearly in the spreadsheet. I can't draw a graph as it is a 4 dimensional figure.

However the minimum value of area occurs at a different pair of values for a,b for every value of

0<r<∞

Notice r cannot be zero or infinity as that would require b or a to be infinity and the other to be zero at the minimum 'area'

This question contains more than meets the eye.
You have provided no new information. All of this was understood by the OP before he even posted his question. The posted replies also addressed all of this again since we didn't realize he had correctly solved the problem that he posted according to his interpretation. Now you have just reiterated it all again in greater detail, since you clearly did not read the responses, or scanned them too quickly. (which is fine, as you may be busy)

The mystery is whether the drawing the OP provided represents a correct interpretation of the problem as quoted from the book. If so, the book answer is wrong. If not, what is the correct interpretation?

I'm not sure if the OP cares anymore as he should realize that he has a good understanding of the material he is studying. Personally, I could just dismiss the problem as a poorly worded question, but also I'm curious if there is another interpretation of the words that leads to the book's numerical answer. Perhaps you can focus your considerable intellect on this latter issue? I ask respectfully. I tried and failed, and if no one else cares to pursue it, then there is not much more to address in this thread.
 

BillO

Joined Nov 24, 2008
992
I actually came up with your exact solution Studiot (L=36/sqrt(R) in my case). That’s why I claimed there is no unique answer. The way the question was worded it asked for the dimensions that produced the minimum area and did not hint at finding a function of R.
 

studiot

Joined Nov 9, 2007
5,002
Actually Steve I did look carefully at the words and other's posts.

I looked at the words to see if JUT had perhaps missed a word in the transcription that would nail r down, but could not see any (for instance the statement that the side path is wider or narrower than 3 feet.)

However if r was badly written, or simply a printer's error in the book, and actually the number 4, then the minimum area occurs at
a = 24
b = 18

This can be seen in my spreadsheet or calculated from my formulae.

I can't find reference to this fact in any other post.

I also proved that r cannot be zero for any finite answer. I don't think this was explicitly stated before although there was much discussion about the case of r = 0 and some (incorrect) solutions based upon this value proposed.

So yes I did add more information. I will leave it as an exercise to the reader to read my spreadsheet and see what would happen if r was some other value (9 springs to mind) than 4.

Several others have posted solid correct facts, I never said or implied otherwise.

Whilst hoping that "the dimensions" will yield a numeric answer, there is no rule that requires this. In fact it is more reasonable to expect a formula (function) given a variable to start with (r).

I have proved, not merely stated, that the solution is a function, given r as a variable. Again this was new information.
 
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steveb

Joined Jul 3, 2008
2,431
However if r was badly written, or simply a printer's error in the book, and actually the number 4, then the minimum area occurs at
a = 24
b = 18

That's an interesting possibility.

So yes I did add more information.
Personally, I didn't think so. This was a relatively easy problem (if diagram and wording are assumed correct) and what little that was not said, was obvious. At least 4 other people solved this in about a minute. Basically, you were just preaching to the choir.
 
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