Suppose that $anxy-$coordinate system $is$ superimposed $on$ a metal plate where $x$ and $y$ are mea$-$

sured $in$ centimeters. The metal plate $is$ heated such that the temperature $in$ degrees Celcius $at$

location $(x,y)is$ given $by$

$T(x,y)=\frac{x^2{y}^3}{12}$

$An$ ant $is$ located $at$ position $(3,2)on$ the plate. The ant $is$ going $to$ move $0\mathrm{.}5cm$ from its current

location.

$(i)$ Approximately where should the ant move $toifit$ wants $to$ have the temperature $of$ the plate

under itself increase $as$ much $as$ possible. Round each coordinate $to$ the nearest $0\mathrm{.}01cm.$

$(ii)$ Assuming the ant moves from $(3,2)to$ the location $in$ part $(i),$ compute the actual change $in$

temperature $T$

$(iii)$ Assuming the ant moves from $(3,2)to$ the location $in$ part $(i),$ compute the corresponding

differential $dT$