# Calculus (minima) problem

#### jut

Joined Aug 25, 2007
224

#### Papabravo

Joined Feb 24, 2006
12,488
1. You need to find an expression for the area of the bed and the walk
2. You need to take the derivative of that expression and set it equal to zero.

#### BillO

Joined Nov 24, 2008
990
There is simply not enough information given to give a unique slution to this problem. Sorry.

#### jut

Joined Aug 25, 2007
224
There is simply not enough information given to give a unique slution to this problem. Sorry.
How do you know?

My calculus book gives an answer in the back of the book.

#### Papabravo

Joined Feb 24, 2006
12,488
What is your expression for the area of the garden and the walk together?

#### Ratch

Joined Mar 20, 2007
1,070
jut,

How do you know?

My calculus book gives an answer in the back of the book.
The flower bed is locked into a rectangular area of 432 sq ft. The ends are specified at 3 feet. The total area is going to be a minimum when R=0. If R >0, then the area will become larger and won't be a minimum anymore. So make the rectangular dimensions of of the flower bed anything whose product is 432, such as 18x24 = 432, and the problem will be satisfied if R=0. Of course lots of other values will work too, such as 9x48=432.

Ratch

#### BillO

Joined Nov 24, 2008
990
Right,

Here is a solution based on the available information, but I doubt it's what is in the book.

Set R=0 then we can postulate the total area as,

A= 2*3*S + S*L, where S*L = 432

Then, if we take the limit as S approaches 0 we get A=432. In this case S is infinitesimal and L is infinite. There is a solution for every imaginable value of R. I doubt this is what the textbook is after. This is why I suggest there is something missing in the statement of the problem. Are there some other constraints on R, S or L?

#### steveb

Joined Jul 3, 2008
2,436
I think that this problem is being misread. The idea is not to change R until the area is minimum, but to change the dimensions of the bed until the area is minimum. The value of R would be specified later, and no one can walk on a walkway with 0 width.

Papabravo has the right approach.

A=432+6*S+864*R/S+12*R where S is the length of the bed

Take derivative with respect to S and set to zero.

I did it quick, but I got S=12*sqrt(R), but I could have made a mistake.

The other dimension would be 36/sqrt(R), if the above is correct.

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#### Ratch

Joined Mar 20, 2007
1,070
steveb,

I think that this problem is being misread. The idea is not to change R until the area is minimum, but to change the dimensions of the bed until the area is minimum. The value of R would be specified later,..
Well, 432 < area < ? . No matter what you come up with for ?, the area will be even less when R=0 .

Ratch

#### steveb

Joined Jul 3, 2008
2,436
steveb,

Well, 432 < area < ? . No matter what you come up with for ?, the area will be even less when R=0 .

Ratch
Well, that's a useless and trivial solution in my opinion, but yes, if the gardener decides he doesn't need to walk completely around the garden, then he may decide to make R=0. However, the intent of the problem is to not worry about R right now.

#### jut

Joined Aug 25, 2007
224
jut,

The flower bed is locked into a rectangular area of 432 sq ft. The ends are specified at 3 feet. The total area is going to be a minimum when R=0. If R >0, then the area will become larger and won't be a minimum anymore. So make the rectangular dimensions of of the flower bed anything whose product is 432, such as 18x24 = 432, and the problem will be satisfied if R=0. Of course lots of other values will work too, such as 9x48=432.

Ratch

I thought about that at first but then abandoned the idea since the problem says there garden is surrounded by a walk. However, setting R=0 does give the answer in the book, which is 18x24. But still, you cannot pick any two numbers whose product is 432 because you have to make the total area a minimum.

If R=0, an expression for the total area is:

LW + 2*3W

where,
W = garden width
L = garden length

To minimize the above equation, you could simply make the width very small and make the length very long. For example...

433 * 0.99769 = 432 Area = 437.98614
10000 * 0.0432 = 432 Area = 432.25920

I wouldn't want a garden that's two miles long by 16th of an inch. But anyways, I just don't think the problem was worded correctly.

#### jut

Joined Aug 25, 2007
224
Right,

Here is a solution based on the available information, but I doubt it's what is in the book.

Set R=0 then we can postulate the total area as,

A= 2*3*S + S*L, where S*L = 432

Then, if we take the limit as S approaches 0 we get A=432. In this case S is infinitesimal and L is infinite. There is a solution for every imaginable value of R. I doubt this is what the textbook is after. This is why I suggest there is something missing in the statement of the problem. Are there some other constraints on R, S or L?
No, I copied the problem verbatim from the book.

#### jut

Joined Aug 25, 2007
224
The other dimension would be 36/sqrt(R), if the above is correct.
That's how I originally solved the problem, by finding the length of the flower bed in terms of r. I also got 36/sqrt(R). I guess the intention was to find the side or length in terms of r, but when the book gives an answer of 18x42 it messed with my head.

BTW everyone, thanks for the help. I posted this problem on a few other math forums and haven't got any replies --I love this forum. #### steveb

Joined Jul 3, 2008
2,436
That's how I originally solved the problem, by finding the length of the flower bed in terms of r. I also got 36/sqrt(R). I guess the intention was to find the side or length in terms of r, but when the book gives an answer of 18x42 it messed with my head.
Yeah, something is mis-worded I think. Is the diagram you provided from the book, or did you draw that yourself based on the wording of the question?

#### jut

Joined Aug 25, 2007
224
Is the diagram you provided from the book, or did you draw that yourself based on the wording of the question?
Yes, I drew that. But I did copy the problem description verbatim.

#### steveb

Joined Jul 3, 2008
2,436
Yes, I drew that. But I did copy the problem description verbatim.
Personally, I wouldn't know how to draw the diagram, with confidence, based on the wording of the problem. It seems too vague to me. Maybe we're all missing an obvious interpretation, but it seems ill-defined to me, and you seem to know what you are doing.

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#### blazedaces

Joined Jul 24, 2008
130
I hate to say this, but the problem is either an incomplete one, or a bad one. If it's exactly written that way, then this is more for a word problem than a math problem. The area of the bed AND walk together will always be a minimum when R = 0 (as stated previously). Creating an equation and taking the derivative is extra, unnecessary work.

Good luck,
-blazed

#### studiot

Joined Nov 9, 2007
4,998
This question has obviously generated some interest over the holiday.
However the Christmas spirit seems to have befuddled some brains.

So let me assure you that this problem is capable of solution, as presented by JUT, and contains all the necessary information to do so.

What is does not ask for is a solution of numbers only.

In fact the solution sides of the flower bed are functions of the path width, r.

I have appended a new diagram as it it easier to use the flower bed sides, denoted a and b.

From this we can obtain an objective function to be minimised, either analytically or by non-linear programming in a 4 dimensional hull.

In the zip file I have tabulated in Excel the garden area for various values of r, a and b.
A valley can be seen to be wandering up from the origin past large b for small r towards large a for large r. I have highlighted the minima in red.

Those who wish to develop their own solution can test that against the data in this spreadsheet.

Alternatively those whose wish to continue to live the easy life can look at the finished analytical solution in my third attachment.

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#### steveb

Joined Jul 3, 2008
2,436
This question has obviously generated some interest over the holiday.
However the Christmas spirit seems to have befuddled some brains.

So let me assure you that this problem is capable of solution, as presented by JUT, and contains all the necessary information to do so.

What is does not ask for is a solution of numbers only.
Actually, the OP already went down this path which turned out to be what a few of us suggested also. The problem is that the book does give a numerical solution as the answer. So, either there is a mistake in the book, or we are misinterpreting the question.

#### studiot

Joined Nov 9, 2007
4,998
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.