Optimization of a rectangular area divided....

[miss_b]

Hello....again.

I have a question, where I believe I think my answers are correct, but the answers on my Webtest say that I am wrong (hope that makes sense ) Anyway:

Q:
A farmer has 1737 meters of fencing and wants to enclose a large rectangular area and divide it into 3 parallel pens. What are the dimensions of the entire rectangle such that the enclosed are is maximised?

[attachment=0:ri55g0j0]pens.jpg[/attachment:ri55g0j0]

\(\displaystyle L = 1737\)

\(\displaystyle L = 6b + 4a = 1737\)

\(\displaystyle a = \frac{1737 - 6b}{4}\)

\(\displaystyle (AREA) A = 3ab\)

\(\displaystyle = 3 \left[\frac{1737 - 6b}{4}\right] b\)

\(\displaystyle = \frac{3}{4} (1737 - 6b^2)\)

Differentiate:
\(\displaystyle \frac{dA}{db} = \frac{3}{4} (1737 - 12b)\)

Maximum A = \(\displaystyle \frac{dA}{db} = 0\)

\(\displaystyle \frac {3}{4} (1737 - 12b) = 0\)

\(\displaystyle b = \frac {1737}{12} = 144.75\)

\(\displaystyle a = \frac {1737 - 6 (144.75)}{4} = 217.125\)

Now, when I plug in the a and b values into the \(\displaystyle L = 6b + 4a\) equation I get 1737 right? But the answers on my Webtest are:
Length of parallels = 217.13 m - YES
Length of end sides = 434.25 m - NO??? (What is going on here??)

Why doesn't this work?

Thank you for your help

Beckie

 

[royhaas]

Your work appears correct, assuming that the three pens share a side. It seems the only other possibility is to have three separate pens, which doesn't make sensse. That would be the same as using 1737/3 m of fencing for one pen, resulting in 3 square pens.

 

[soroban]

Hello, Beckie!

A farmer has 1737 meters of fencing and wants to enclose a large rectangular area and divide it into 3 parallel pens.
What are the dimensions of the entire rectangle such that the enclosed are is maximised?

\(\displaystyle b \:=\: \frac {1737}{12} \:=\: 144.75\)

\(\displaystyle a \:=\: \frac {1737 - 6 (144.75)}{4} \:=\: 217.125\)

But the answers on my Webtest are:
. . Length of parallels = 217.13 m
. . Length of end sides = 434.25 m
What is going on here?

Your work is correct, but you didn't answer the question.

What are the "dimensions of the entire rectangle"?

Look at the diagram.
The rectangle is 3b meters long and a meters wide.

 

[miss_b]

Hey Soroban,

Look at the diagram.
The rectangle is 3b meters long and a meters wide.

Who would have deduced that my crappy 'MS Paint' drawing would have been my own personal nemesis? He he he.

Thank you so much for pointing out the obvious to me LOL. I was sweating on the question for hours, maybe not enough or too much caffiene...

 

[mmm4444bot]

\(\displaystyle \frac{3}{4} (1737 - 6b^2)\)

This expression is wrong.

 

[Deleted member 4993]

Hello....again.

I have a question, where I believe I think my answers are correct, but the answers on my Webtest say that I am wrong (hope that makes sense ) Anyway:

Q:
A farmer has 1737 meters of fencing and wants to enclose a large rectangular area and divide it into 3 parallel pens. What are the dimensions of the entire rectangle such that the enclosed are is maximised?

[attachment=0:2kxiw7sa]pens.jpg[/attachment:2kxiw7sa]


\(\displaystyle = \frac{3}{4} (1737b - 6b^2)\) ..............missed the 'b' - but rest is OK

Differentiate:
\(\displaystyle \frac{dA}{db} = \frac{3}{4} (1737 - 12b)\)

Maximum A = \(\displaystyle \frac{dA}{db} = 0\)

\(\displaystyle \frac {3}{4} (1737 - 12b) = 0\)

\(\displaystyle b = \frac {1737}{12} = 144.75\)

\(\displaystyle a = \frac {1737 - 6 (144.75)}{4} = 217.125\)

Now, when I plug in the a and b values into the \(\displaystyle L = 6b + 4a\) equation I get 1737 right? But the answers on my Webtest are:
Length of parallels = 217.13 m - YES
Length of end sides = 434.25 m = 3 * 144.75 - YES - NO??? (What is going on here??)

Why doesn't this work?

Thank you for your help

Beckie

 

[miss_b]

\(\displaystyle = \frac{3}{4} (1737 - 6b^2)\)

I didn't even think to apply 'b' across all. My lecturer missed the 'b' in this equation in the lecture example too.

Thank you!!

 

[mmm4444bot]

I didn't even think to apply 'b' across all. My lecturer missed the 'b' in this equation in the lecture example too.

Then, perhaps, we should rename both of you: missed_b . :lol:

 

[miss_b]

LMAO!!!!