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 an initial value of ${t}_{0}=0.5$, to find the number of months after Divid-e20 was discovered that R first equals 1.

Give your answer to 3 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, calculate 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.

Explain how to deduce this value from the model, $R(t)$.