6 Helga invests \(\pounds 4000\) in a savings account.
After \(t\) days, her investment is worth \(\pounds y\).
The rate of increase of \(y\) is \(k y\), where \(k\) is a constant.
- Write down a differential equation in terms of \(t , y\) and \(k\).
- Solve your differential equation to find the value of Helga's investment after \(t\) days. Give your answer in terms of \(k\) and \(t\).
It is given that \(k = \frac { 1 } { 365 } \ln \left( 1 + \frac { r } { 100 } \right)\) where \(r \%\) is the rate of interest per annum. During the first year the rate of interest is \(6 \%\) per annum.
- Find the value of Helga's investment after 90 days.
After one year (365 days), the rate of interest drops to 5\% per annum.
- Find the total time that it will take for Helga's investment to double in value.