Maxima question needs checking

[JakePearson]

question

a rectangular plot of land requires fencing on all 4 sides. two opposite sides will use heavy duty fencing at £6/metre, whilst the other tow sides will use standard fencing at £3/metre. if the farmer buying the fencing has £5000 to spend, what is the maximum area he can enclose?

attempt

let y = heavy duty fencing sides and x= standard fencing sides.

A=x*y
5000=6y+6y+3x+3x
5000=12y+6x...solve for x (or y)...
5000-12y=6x (divide by 6)...
833.33-2y=x...Now we will substitute this into A=x*y

A=(833.33-2y)*y
A=833.33y-2y² ...now take derivative and set equal to 0
A'=833.33-4y=0
or...4y=833.33
y=208.33 meters.find x
5000-(12*208.33)=6x
2500=6x
x=416.66 meters

So max area is x*y or (208.33*416.66)= 86803m².

 

[Gib Z]

That is the correct method and solution. It may be slightly better to use the exact values, eg 2500/3, rather than 833.33, and also instead of differentiating and solving, you could use the formula for the axis of symmetry of a parabola x=-b/2a, might be quicker next time.

 

[Architeuthis Dux]

I would suggest that the solution provided is correct and the maximum area that can be enclosed is 86803m2. However, I would also recommend that the assumptions and calculations made in this solution should be checked and validated by conducting further research and analysis. Additionally, it would be important to consider any potential limitations or factors that may affect the actual implementation of this solution, such as the availability of fencing materials or the terrain of the land. Overall, this solution provides a good starting point for determining the maximum area that can be enclosed within the given budget, but further investigation and consideration may be necessary for a more accurate and reliable result.