Functions of Several Variables — Question 4

Question 4 – Solution

Strategy Take natural logarithms after isolating the exponential.

Step 1: General level From $T_0/100=e^{-r^2/4}$, $$\ln(T_0/100)=-r^2/4\Longrightarrow\boxed{r^2=4\ln(100/T_0)}.$$ Thus levels are concentric spheres; $T_0=100$ degenerates to the origin.

Step 2: Fifty-degree surface For $T_0=50$, $r^2=4\ln 2$, so $\boxed{r=2\sqrt{\ln 2}}$.

Step 3: Path On $(t,t,0)$, $r^2=2t^2$. Hence $$2t^2=4\ln 2\Longrightarrow t=\pm\sqrt{2\ln 2}.$$ Starting at $t=0$ and moving with increasing $t$, the first time is $\boxed{t=\sqrt{2\ln 2}}$.