An unknown amount of pollutant is dropped in a pond with a volume of approximately 200 m³.  A flow rate (Q) of 10 m³ / minute flows into and out of the pond after the initial pollution event. The rate of change of concentration (C) of pollutant in the pond is given by:

\normalsize\frac{dC}{dt} = -\frac{Q}{V} C

After 30 minutes the concentration of pollutant in the pond is found to be 0.01 kg/m³.  Using an initial guess of 0.1 kg/m³ and the Euler method with 50 steps, apply Fixed Point Iteration with a convergence of less than 0.0001 between iterations to find the initial concentration of pollutant in the pond.

Use the Newton-Raphson method with an initial guess of 5 to find a root of the following equation to a convergence of less than 0.001:

\normalsize \textrm{e}^{t} - 5t^2 + t + 5 = 0

When you substitute your found root into the original equation, what value do you calculate?

Use the Secant method with initial guesses of 0 and 0.5 to find a root of the following equation to a convergence of less than 0.001:

\normalsize t^3 + 2t^2 - 5t + 1 = 0

When you substitute your found root into the original equation, what value do you calculate?

A cylindrical tank, with the circular ends parallel to the ground, is leaking through a circular hole in the bottom.  The tank has a diameter of 4 m, and the hole has a diameter of 0.1 m.

\normalsize\frac{dh}{dt}=-9\frac{A_1}{A_2}\sqrt{h} 

where h is the height of water in the tank, t is time, A₁ is the cross sectional area of the hole and A₂ is the cross-sectional area of the tank.

The initial height of water in the tank is 1.5 m.  

What is the error at a time of 200 seconds between the analytical solution and a fourth order Runge Kutta method of 5 steps

A ceramic component is removed from a firing oven at a 1000 K, and left to cool in a temperature controlled environment of 295 K.

The component cools according to the relationship shown in the following equation:

\normalsize\frac{dT}{dt} = k (T - T_a) 

where T is the temperature of the component, t is time, k is a constant with value of -0.0185, and T_a is the temperature of the temperature controlled environment.

What is the error between the analytical solution and a 50 step Euler method at 60 minutes?