Functions of Several Variables — Question 4

Question 4 – Solution

Strategy Set $T$ equal to a constant and take logarithms.

Isotherm Setting $T=50$ gives $e^{-(x^2+4y^2)/8}=1/2$, hence $$x^2+4y^2=8\ln 2.$$ The semiaxes are $\sqrt{8\ln 2}$ and $\sqrt{2\ln 2}$, so the area is $\boxed{4\pi\ln 2}$.

Maximum and spacing The maximum is $100^\circ$ at $(0,0)$. For level $0<c<100$, $x^2+4y^2=8\ln(100/c)$; lower temperatures correspond to larger ellipses, with nonlinear spacing caused by exponential decay.