Scowart

My team matrix: [

  [0, 1, 1, 3, 1],

[2, 0, 2, 2, 2],

[3, 4, 0, 1, 4],

[1, 4, 1, 0, 2],

[4, 3, 2, 3, 0]

]

The ranking for the league was:

Team 3: rating = 0.5616064344284055
Team 5: rating = 0.5279808500356232
Team 2: rating = 0.4002177984108782
Team 4: rating = 0.3946242396754421
Team 1: rating = 0.29988640866168415

Team 3 was the best.

Based on my assigned parameters, the temperature in the lower right corner of the bar after reaching steady-state is 0.90877.

Given my specific domain and parameters, the temperature at $t=1s$ near the midpoint of the insulated edge of the triangle is $0.57228$.

A metal bar of length 1 lies along the unit interval. Its temperature distribution is given by

$$g(x) = 5x^2 - 1x.$$

At time $t=0$, its left end is set to temperature $0$ and its right end to $-3$. Sketch the temperature distribution at times

$$t=0, t=0.01, t=0.1, t=10.$$

The heat flow equation with a constant internal heat source I was given is

$$u_t=u_{xx}+5.$$

My boundary conditions are

$$u(0,t)=-4\mbox{ and }u(1,t)=5.$$

To find the steady state temperature distribution we set the partial derivative of the function $u(x,t)$ with respect to time equal to zero, that is $u_t=0$, and then integrate in succession, yielding

$$u(x,t)=-\frac{5}{2}x^2+c_1x+c_2.$$

Using the given boundary conditions, $u(0,t)=-4$, $u(1,t)=5$, we determine $c_1=\frac{23}{2}$ and $c_2=-4$. Thus the steady state temperature distribution is represented by

$$u(x,t)=-\frac{5}{2}x^2+\frac{23}{2}x-4.$$