Solving for the maximum value of x and y

A

allan_89_04

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Apr 1, 2007
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Hi,
I have been working on this question in my exam practice handouts for literally hours...can someone plz help me out on how i can work out the solution for this? cause the questions i dont know how do do always seems to pop up in exams

A farmer wishes to fence each of Nine identically shaped fields (all of length x m and width y m) as indicated below, useing a stream to save fencing material along one side of the 4 fields. If there is 9km of fencing material available, what dimensions (to the nearest meter) should each field be so the total area enclosed is a maximum? What is this maximum area?
(HINT: you might find it easier to solve y rateher then y to start with)

(This question is worth [8] marks if it helps at all]
 
I don't know what the fenced area is supposed to look like. You didn't include the diagram, but here's a go.

12y+9x=9000..[1]
A=9xy...[2]

Solve [1] for y and sub into [2]:

\(\displaystyle \L\\A=6750x-\frac{27}{4}x^{2}\)

\(\displaystyle \L\\A'=6750-\frac{27}{2}x\)

Set to 0 and solve for x, we get x=500 and y=375.
 
galactus said:
I don't know what the fenced area is supposed to look like.
Click to expand...
sorry about that, i forgot to add it. it looks like this on the sheet:
.....................................x
....................._________________________
...................y|____________|___________ |_____________
. ___________ |____________|___________ |____________|
.|..................|.....................|.....................|......................|
Stream is here so no fencing is require at the bottom

so basically there are 12 y values, and 9 x values and 9 fields.
The diagram looks ok when i type it out, but comes all stuffed up when it submits
 
\(\displaystyle \L\\A=6750x-\frac{27}{4}x^{2}\)
does anyone know where he got the \(\displaystyle 27/4\) from?
 
I solved [1] for y and subbed into [2]. Then differentiated.
 
From [1]:

\(\displaystyle \L\\y=12y+9x=9000\)

Solve for y:

\(\displaystyle \L\\y=750-\frac{3}{4}x\)

Sub into [2]:

\(\displaystyle \L\\9x(750-\frac{3}{4}x)\)

Distribute:

\(\displaystyle \L\\6750x-\frac{27}{4}x^{2}\)

Now differentiate.
 
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