Solving Optimization Problems AB & BC

Solving means differentiating the reduced objective, solving for critical numbers, keeping only those in the domain, justifying that one is the extremum, and back-substituting. For $1200$ feet of fence on three sides, $A'(x) = 1200 - 4x$ gives $x = 300$, $A'' = -4$ confirms a maximum, $y = 600$ follows from the constraint, and the maximum area is $180{,}000$ square feet. For the can, $4\pi r^{3} = 710$ gives $r \approx 3.837$ centimetres with $h \approx 7.675$, and $S'' > 0$ throughout $r > 0$ makes the single critical point absolute.

The dominant error is reporting the wrong quantity. A solved fence problem contains $x = 300$, $y = 600$, the pair of dimensions, and the area $180{,}000$, all of them true, and only one answers any given prompt; the critical input is the most tempting and the least often asked for. Two structural results are worth knowing and are over-applied: a fixed perimeter on four sides is best as a square, which the three-sided fence problem does not obey, and a closed cylinder of fixed volume has $h = 2r$, which an open-topped one does not.