The $C{O}_2$ concentration in the atmosphere at a particular point on earth follows a cyclic pattern, due

to seasonal changes, in particular in the relative proportions of photosynthesis and respiration in plants

at different months of the year. In order to explore this cycle, we can decompose the net flux of $C{O}_2$

as

$\frac{dc}{dt}=i(t)-o(t)$

where $i(t)$ is the rate of carbon influx into the atmosphere due to respiration and emissions and $o(t)$ is

the rate of carbon outflux out of the atmosphere due to photosynthesis. Both are given in $Pg\frac{C}{?}$ year

$($Petagrams of carbon per year, where $1$ petagram $={10}^{15}$ grams$).$

Let $a0$ and $b>$a represent two different points in time.

$($b$)**{*}^{{?}^{?}}$ Describe concisely the physical quantity that is represented by ${\displaystyle \underset{a}{\overset{b}{}}}(i(t)-o(t))dt.$

Suppose we let the start of the year $($January $1$st$)$ be time $t=0$ and start measuring the change in

$C{O}_2$ at this time. Assume that $i(t)=25+5cos(2t)$ and that $o(t)=25(1+cos(2t-)).$

$($a$)*********$ Compute the earliest time $T>0$ at which the amount of $C{O}_2$ in the atmosphere

starts decreasing

$($b$)*********$ Does the amount of $C{O}_2$ in the atmosphere return to its original value? If so$,$

determine the earliest time $t^{\text{'}}>0$ at which this occurs.

On climatological time scales, the annual oscillations of $C{O}_2$ discussed above are small compared

to the overall increase in carbon. One possible way to determine the overall increase in carbon is

by using averages.

For a function $f(t),$ its average value from $t=a$ to $t=b>a$ is given by

$\frac{1}{b-a}{\displaystyle \underset{a}{\overset{b}{}}}f(t)dt$

Below is a plot of the $C{O}_2$ concentration at Mauna Loa Observatory since $1960.$ By observation a

simple model of the concentration of atmospheric $C{O}_2$ since $1960$ is

$C(t)=310+4cos(2t)+1\mathrm{.}65t$

where $t$ is the time in years after $1960$ and here $C(t)$ is measured in ppm $($parts per million$).$ The

long$-$term trend of rising carbon dioxide levels is driven by human activities.

Let $A_v(t)$ be the yearly average of the concentration, namely

$A_v(t)={\displaystyle \underset{t}{\overset{t+1}{}}}C(s)ds.$

$*********$ For tin$[0,1],$ find the time at which the yearly average is maximal.

$*********$ Calculate the rate of increase of $A_v(t).$ That is$,$ how much is the yearly average concen$-$

tration of atmospheric $C{O}_2$ increasing per year?