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.