A person decides in 2020 that they want to completely eradicate their carbon footprint in 20 months.

Following this decision, they begin to use multiple technologies that decrease their carbon footprint, such as limiting air travel, carbon offsetting and a solar powered home that contributes energy back to the national grid.

This results in their rate of emissions per month, E, following the equation

\[E = 12\ln\left( t + 5 \right) – 2t + 2\]

where *t* is the time, in months, since 2020.

a) Show that they achieve zero emissions between 18 and 22 months after they start.

[2 marks]

b) Using the iteration formula \(t_{n + 1} = 6\ln{(t + 5)} + 1\) with \(t_{o} = 18\), find the value of \(t\) at which they achieve zero emissions to 2 decimal places.

[3 marks]