Optimal Garden Shape for Minimizing Fencing

[ducmod]

Homework Statement

Hello!
It seems that I don't know the correct approach to this problem.

Sally wants to plant a vegetable garden along the side of her home. She doesn't have any fencing, but wants to keep the size of the garden to 100 square feet.
a) What are the dimensions of the garden which will minimize
the amount of fencing she needs to buy?
b) What is the minimum amount of fencing she needs
to buy? Round your answers to the nearest foot. (Note: Since one side of the garden will
border the house, Sally doesn't need fencing along that side.)

Homework Equations

My thoughts:
x - width of the garden
l - length
Since one wall is not required, the fencing = 2x + l
area = x*l = 100, hence l = 100/x

Have no idea how to proceed to solve a) and b) (without calculus, based only on precalculus program).

Thank you!

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

Hello!
It seems that I don't know the correct approach to this problem.

Sally wants to plant a vegetable garden along the side of her home. She doesn't have any fencing, but wants to keep the size of the garden to 100 square feet.
a) What are the dimensions of the garden which will minimize
the amount of fencing she needs to buy?
b) What is the minimum amount of fencing she needs
to buy? Round your answers to the nearest foot. (Note: Since one side of the garden will
border the house, Sally doesn't need fencing along that side.)

Homework Equations

My thoughts:
x - width of the garden
l - length
Since one wall is not required, the fencing = 2x + l
area = x*l = 100, hence l = 100/x

Have no idea how to proceed to solve a) and b) (without calculus, based only on precalculus program).

Thank you!

The Attempt at a Solution

So, you want to minimize the fencing length $L(x) = 2x + 100/x$ over $x > 0$. One way is to plot a graph of the function and see where the minimum occurs. This would at least give you an approximate estimate of where the minimum occurs.

Another way (without calculus) is to use the arithmetic/geometric inequality, which says that for $a, b > 0$ we have
$$\frac{a+b}{2} \geq \sqrt{ab},$$
with equality holding if, and only if $a = b$. Sometimes this can give a slick way of minimizing a sum of two terms, such as you have in your problem.

 

[BvU]

I Sally forced to lay out a rectangular garden ?

 

[RUber]

Homework Equations

My thoughts:
x - width of the garden
l - length
Since one wall is not required, the fencing = 2x + l
area = x*l = 100, hence l = 100/x

First put the equation into one variable.
$F = 2x + 100/x$
Then you can multiply by x to get:
$2x^2 - Fx + 100 = 0$
Find possible solutions to the quadratic equation in terms of F, and you will see what the minimum possible F is because of the radical involved. When you have F, work backward to find x and l.
Like Ray mentioned, a quick plot will at least put you in the ball park, so you know if your solution makes sense.

 

[ducmod]

Thank you very much for your help!

 

[BvU]

well, what did you find ? And is it shorter than a half-circle fence ?

 

[haruspex]

well, what did you find ? And is it shorter than a half-circle fence ?

You make a good point, but the question does ask for dimensions, plural, and does not ask about shape. So I suspect the questioner intended rectangular.
On that assumption, there's a way which does not even use algebra, provided you assume that a square would be the ideal rectangle if there were no house wall to make use of.

 

[NickAtNight]

1) How many equations do you have? You said: Fence = 2x+l and x*l=100
2) How many unknowns (or variables) do you have?

 

[NickAtNight]

Have no idea how to proceed

INITIAL GUESS:
When you have a problem and you do not know how to solve it, one approach is to make an initial guess.

Example:
low guess: The width of the garden plot will be 1 foot.
apply your first formula. Then the length must be l = 100/w = 100 feet
... boy does sally have a long house !
apply your second formula. Then the amount of fence needed must be Fence = 2x+l = 2*1+100 = 102
put your result in a table
... Width Length Fence
... 1 100 102

MAKE A SECOND GUESS:
Increment: 1 foot
New width is 2 feet
Apply the first formula. l = 100/w = 50 feet
Apply your second formula. Fence = 2x+l = 2*2+50 = 4+50 = 54
... Width Length Fence
... 1 100 102
... 2 50 54

REVIEW THE RESULT
Which answer is better?
Move in the direction that is better.
In this case, 2 feet is a lot better than 1 feet.
So now try 3 feet.

CONTINUE AND REPEAT THE METHOD

 

[insightful]

A semi-circle is the logical answer. The problem could hardly have been stated, "What is the dimension of the garden..."

 

[haruspex]

A semi-circle is the logical answer. The problem could hardly have been stated, "What is the dimension of the garden..."

No, but it would have asked for shape.

 

[NickAtNight]

A semi-circle is the logical answer. "

Only mathematically. Most gardens are rectangular.

 

[insightful]

Only mathematically.

Note that this is not a gardening forum.

 

[insightful]

but the question does ask for dimensions, plural, and does not ask about shape.

How about the dimensions of a trapezoid (or better yet, a trapezoid next to a rectangle)?

 

[haruspex]

How about the dimensions of a trapezoid (or better yet, a trapezoid next to a rectangle)?

I repeat, it does not ask about shape, suggesting the shape is assumed.

 

[insightful]

I repeat, it does not ask about shape, suggesting the shape is assumed.

I agree; the OP must assume a shape to get the minimum fence length.

 

[haruspex]

I agree; the OP must assume a shape to get the minimum fence length.

No, you are saying the OP must choose a shape. As I'm sure you understand, I am arguing that the problem setter has assumed a specific shape (namely, a rectangle) and has failed to realize it was not made explicit.

 

[insightful]

I am arguing that the problem setter has assumed a specific shape (namely, a rectangle) and has failed to realize it was not made explicit.

So, the correct answer to the instructor is:

Assuming you meant a rectangle, it measures 7'X14' requiring 28' of fencing.
Assuming you meant what you wrote, it's an 8' radius semicircle requiring 25' of fencing.

 

[Ray Vickson]

So, the correct answer to the instructor is:

Assuming you meant a rectangle, it measures 7'X14' requiring 28' of fencing.
Assuming you meant what you wrote, it's an 8' radius semicircle requiring 25' of fencing.

There are two problems with your posting:
(1) Since the OP has not told us his answer (and we don't know if he did correct work) we should not supply give complete answers for him. PF rules require we give hints only, not answers, at least not until the work has already been turned in for marking, etc.
(2) Hinting to the instructor his/her failings is likely a bad move on the student's part. Better to leave out the "Assuming you meant..." parts, and just say: "Assuming a rectangular shape..." and/or "Assuming a semi-circular shape ..." . (Even then, proving that the optimal shape is a semi-circle involves constrained calculus of variations, which goes way beyond pre-calculus. Of course, it is intuitive, but intuition is not proof.)

 

[insightful]

Sorry, didn't realize PF is a humor-free zone.

;-]

 

[Mark44]

A semi-circle is the logical answer. The problem could hardly have been stated, "What is the dimension of the garden..."

? "could hardly have been stated" ? I don't get your point here.

The problem statement was, "What are the dimensions of the garden..." so how can inferring that the shape of the garden was semicircular be logical? The most reasonable assumption, and one that I'm almost certain was intended by the problem author, is that the garden is rectangular in shape. As haruspex points out, if the intent was to work with a garden that was semicircular, trapezoidal, pentagonal, or any shape other than rectangular, that information would have been given in the problem.

Sorry, didn't realize PF is a humor-free zone.
;-]

It's not. Was there some humor in this thread?

 

[haruspex]

Sorry, didn't realize PF is a humor-free zone.

;-]

No, it's not that. Perhaps your post #18 was intended tongue-in-cheek, which is fine, but there's a risk the OP would take it literally, which Ray is suggesting would not be fruitful. That said, I feel your proposed way of answering the question could be worded so as to have less implied criticism of the question setter.

 

[insightful]

Was there some humor in this thread?

Yes.

;-]

 

[NickAtNight]

I repeat, it does not ask about shape, suggesting the shape is assumed.

Well, if you were to survey 1,000 gardens of the type where one edge of the garden is along the owners house wall, what would be the distribution of the different types of shapes that you would expect?

Top five answers on the board.

Stick with the most probable shapes to avoid annoying the questioner.

 

[haruspex]

Well, if you were to survey 1,000 gardens of the type where one edge of the garden is along the owners house wall, what would be the distribution of the different types of shapes that you would expect?

Top five answers on the board.

Stick with the most probable shapes to avoid annoying the questioner.

Would you agree that the question is ambiguous in that although it does not specify a rectangular shape there are reasons to suspect that is what the questioner has in mind? That is all I am claiming.