1.1. Scientists are analysing the infection rate, R, at which a virus Divid-e20 spreads amongst a population.They model R using the function

  $R(t)=0.5+{e}^{-0.3t}sin(t)(t>0)$

where t is the number of months since the first case of the virus was discovered.

To control the spread of Divid-e20 scientists need to keep the value of R below 1.

The diagram below shows the graphs of $y=R(t)$ and $R=1$.

Use the Newton-Raphson method, with a suitable initial value, to find the number of months after Divid-e20 was discovered that R first equals 1. Give your answer to three decimal places.

1.2. Given that the infection rate for Divid-e20 first went above 1 on 1^st July, and drops back below 1 roughly 1.997 months (to three decimal places) after Divid-e20 was first discovered, work out the date when the infection rate drops back below 1.

1.3. In the long term, scientists expect the value of R to remain roughly constant around 0.5. Briefly explain how you can tell this from the function R(t).