The methane production in a landfill is based on the equation shown below. The methane generated from the landfill is harnessed for electricity generation. Over a period of five years, the landfill receives an annual amount of $180,000$ tons of waste before it discontinues accepting any additional waste. The landfill gas emission constant is $0\mathrm{.}25$ per year, and the methane generation potential of the waste is $1,000\frac{m^3}{?}$ ton.

Calculate the following:

$($a$)$ With a landfill lifespan of $20$ years, what is the total cumulative methane production $($in

$\{$:$m^3)?(15$ points$)$

$($b$)$ If the overall lifespan of the landfill is extended to $40$ years, what is the percentage

increase in cumulative methane production compared to a $20-$year lifespan? Is it

beneficial, in terms of energy production, to extend the lifespan of the landfill? $(15$

points$)$

Note: Please show the $($i$)$ gas production rate and $($ii$)$ cumulative gas produced lines for $40$

years on a figure.

where

$Q_T=$ total gas emission rate from a landfill $($volume$/$time$)$

$n=$ total time period waste is in a landfill

$k=$ landfill gas emission constant time ${?}^{-1})$

$L_0=$ methane generation potential $($volume$/$mass of waste$)$

$t_i=$ age of the $i^{th}$ section of waste $($time$)$

$M_i=$ mass of waste placed at time $i$