Methane is generated via the anaerobic decomposition (biological degradation in the absence of oxygen) of solid waste in landfills. Collecting the methane for use as a fuel rather than allowing it to disperse into the atmosphere provides a useful supplement to natural gas as an energy source. If a batch of waste with mass M (tonnes) is deposited in a landfill at t = 0, the rate of methane generation at time t is given by V_CH_4 (t) = kL_0 M_waste e^-kt where V_CH_4 is the rate at which methane is generated in standard cubic meters per year, k is a rate constant. L_0 is the total potential yield of landfill gas in standard cubic meters per tonne of waste, and M_waste is the tonnes of waste in the landfill at t = 0. A. Starting with the equation given above, derive an expression for the mass generation rate of methane, M_CH_4 (t). Without doing any calculations, sketch the shape of a plot of M_CH_4 versus t from t = 0 to t = 3 y, and graphically show on the plot the total masses of methane generated in Years 1, 2, and 3. Then, derive an expression for M_CH_4(t), the total mass of methane (tonnes) generated from t = 0 to a time t. B. A new landfill has a yield potential L_0 = 100 SCM CH_4/tonne waste and a rate constant k = 0.04 y^1. At the beginning of the first year, 48,000 tonnes of waste are deposited in the landfill. Calculate the tonnes of methane generated from this deposit over a three-year period. C. A colleague solving the problem of Part (b) calculates the methane produced in three years from the 4.8 times 10^4 tonnes of waste as M_CH_4 (t = 3) = M_CH_4(t = 0) times 1y + M_CH_4 (t = 1) times 1y + M_CH_4 (t = 2) times 1y where M_CH_4 is the first expression derived in Part (a). Briefly state what has been assumed about the rate of methane generation. Calculate the value determined with this method and the percentage error in the calculation. Show graphically what the calculated value corresponds to on another sketch of M_CH_4 versus t. d. D. The following amounts of waste are deposited in the landfill on January 1 in each of three consecutive years. Calculate the metric tons of CH_4 generated through the end of the third year.