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__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 in to A=x*y

A=(833.33-2y)*y

A=833.33y-2y

^{2}....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)= 86803m

^{2}.