A long metallic bar is carrying heat from one of its ends to the other end under steady-state. The variation of temperature $$\theta$$ along the length $$x$$ of the bar from its hot end is best described by which of the following figure.

The metallic bar is assumed to be long and thin so that heat conduction takes place essentially along its length, while simultaneous heat loss occurs from the lateral surface to the surroundings by convection/radiation. This situation is identical to the analysis of a rectangular or cylindrical fin operating under steady state.

Let
  $$T(x)$$ = absolute temperature at a distance $$x$$ from the hot end,
  $$T_\infty$$ = temperature of the surrounding fluid,
  $$\theta(x)=T(x)-T_\infty$$ = excess temperature over the surroundings.

Energy balance on an element of length $$dx$$ gives the one-dimensional fin equation

$$\frac{d^2\theta}{dx^2}=m^2\,\theta \quad -(1)$$
where $$m=\sqrt{\dfrac{hP}{kA}}$$, with
  $$h$$ = convection heat-transfer coefficient on the lateral surface,
  $$P$$ = perimeter of the bar that is exposed to the fluid,
  $$A$$ = cross-sectional area of the bar,
  $$k$$ = thermal conductivity of the bar material.

The general solution of the second-order linear differential equation $$-(1)$$ is

$$\theta(x)=C_1e^{mx}+C_2e^{-mx} \quad -(2)$$

For a very long bar the temperature must remain finite as $$x\to\infty$$; the term $$e^{mx}$$ would diverge, so we set $$C_1=0$$. Hence

$$\theta(x)=\theta_0\,e^{-mx}, \quad \text{where } \theta_0=\theta(0)=T(0)-T_\infty \quad -(3)$$

Equation $$-(3)$$ shows that the excess temperature decays exponentially from the hot end toward the cold end, approaching the ambient temperature asymptotically.

When the bar is plotted with temperature ordinate $$\theta$$ (or $$T$$) against axial distance $$x$$, the curve starts at the highest temperature and gradually flattens as it moves toward $$T_\infty$$, never actually touching it. Among the given sketches, the only one representing such an exponential decay is Option B.

Final Answer: Option B which is: exponential fall of temperature from the hot end that approaches the ambient value asymptotically.