Find the dimensions $of$ a rectangular box with a closed top that has a maximum volume and a surface area $of96$ square feet.

$ax,y,$ and $z.$ Write the general formulas for the volume $V$ and surface area $S.$

$V(x,y,z)=$

$S(x,y,z)=$

$bV(x,y)as$ a function $of$ just two variables, $x$ and $y.$

$V(x,y)=$

$cx>0$ and $y>0in$ this problem.

When optimizing, why must $we$ have $x>0$ and $y>0?$

$A.Itis$ a mathematical convention, but not strictly necessary.

$B.$ The partial derivative formulas are only defined for positive $x$ and $y.$

$C.$ The surface area constraint $is$ only valid for positive $x$ and $y.$

$D.$ Physical dimensions like length and width cannot $be$ zero $or$ negative $ifwe$ want a positive, non$-$zero volume.

$dx,yV(x,y)$ that corresponds $to$ the maximum volume.

Critical point:

$e(x,y,z)(1,2)or(3,4,5).$