The Chain Rule & Implicit Differentiation · Hard problem

A heated plate has temperature $T(x,y)=xy$ (in degrees) at position $(x,y)$ in cm. A sensor moves along the spiral path $r(t)=t+1$, $\theta(t)=\dfrac{\pi t}{2}$ (polar coordinates, $t$ in seconds). Using the chain rule to find $\dfrac{dT}{dt}$ at $t=1$, estimate how many seconds it would take for the sensed temperature to drop by $1$ degree at that instantaneous rate.