= Solution
{c}
Let $A$ be the constant cross-sectional <area>, and suppose that the <temperature> is uniform on each cross-section. Consider a slice $[a,b]$. Its stored thermal <energy> is $\rho cA\int_a^bT(x,t)\,dx$. Since the lateral surface is insulated and there is no internal <heat> generation, <conservation of energy> gives
$$
\rho cA\frac{d}{dt}\int_a^bT(x,t)\,dx=Aq(a,t)-Aq(b,t),
$$
where $q$ is the <heat flux> in the positive $x$ direction. By <Fourier's law>, $q=-kT_x$, so
$$
\rho c\int_a^bT_t\,dx=kT_x(b,t)-kT_x(a,t)=k\int_a^bT_{xx}\,dx.
$$
The slice is arbitrary; therefore the local <heat equation> is
$$
\boxed{T_t=\kappa T_{xx},\qquad\kappa=\frac{k}{\rho c}>0.}
$$
The constant $\kappa$ is the <thermal diffusivity>. The cancellation of $A$ uses a rod of uniform cross-section; the one-dimensional model assumes no transverse <temperature> variation.
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