The ends of a rod of length L are insulated, but heat escapes through the lateral surface of the rod into the surrounding medium, which is at temperature 50. The heat travels through the boundary surface much slower than it travels down the rod, so in any cross section of the rod the difference between the temperature at the center and the temperature at the edge is negligible. The initial temperature is 100 throughout the rod.
Choose the PDE and boundary/initial conditions that model this scenario.

Partial Differential Equation

A. $\displaystyle\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2}=0, \quad0<x<L,0<y<b$

B. $\displaystyle\alpha^2\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}, \quad 0<x<L, \quad t>0$

C. $\displaystyle k\frac{\partial^2 u}{\partial x^2} -c(u-50) = \frac{\partial u}{\partial t}, \quad 0<x<L, \quad t>0$


Boundary/Initial Conditions

A. $\begin{aligned} \frac{\partial u}{\partial x}\Bigg\vert_{x=0} & = 0, &\frac{\partial u}{\partial x}\Bigg\vert_{x=a} &= 0, &0 < t \\ u(x,0) &= 100, & &0 < x < a\\ &&&&\\ &&&&\\ \end{aligned}$
B. $\begin{aligned} u(0,t) &= 0 &u(L,t) &= 0, &t > 0\\ u(x,0) & = 100,&\frac{\partial u}{\partial t}\Bigg\vert_{t=0} &= 0, &0 < x < L\\ &&&&\\ &&&&\\ \end{aligned}$
C. $\begin{aligned} u(0,y) &= 0 &u(L,y) &= 0, &0<y< b\\ u(x,0) & = 100,&& &0 < x < L\\ &&&&\\ &&&&\\ \end{aligned}$