A company's production starts off with initial daily production of $1800$ metric tons and

builds up exponentially at an annual rate of $10\%$ to a peak where the production

rate is thrice the initial rate of production. The production continues at this peak

for $5$ years, and then declines exponentially at an annual rate of $15\%$ for $7$ years

before production was abandoned. Estimate:

$($i$)$ The instantaneous build up rate and the production rate at the end of the

build$-$up$.$

$($ii$)$ The time taken to complete the build$-$up phase.

$($iii$)$ The cumulative production in the build$-$up phase.

$($iv$)$ The cumulative production in the plateau phase.

$($v$)$ The instantaneous decline rate and the production rate at the time of

abandonment.

$($vi$)$ The cumulative production in the decline phase.

$($vii$)$ The PV$($Production$)$ in the decline phase assuming $10\%$ discount rate.

$($viii$)$ If the entire production phases are replaced with a decline phase, how long

will the production take place to have one$-$fifth the cumulative production

as that given by the three phases?