At the start of \(2026\), Lavinia receives a gift of \(\$15\,000\). She wants to buy a boat which costs \(\$28\,000\) so she decides to invest her money.
On \(1\) January \(2026\) Lavinia invests her money in a bank account which pays interest at a nominal annual rate of \(4.8\%\), compounded quarterly. The interest is paid into her account on the last day of each quarter. She makes no further deposits to, or withdrawals from, the account.
Find the amount of money Lavinia will have in her bank account on \(1\) January \(2031\). Give your answer correct to the nearest dollar.[3]
Show that Lavinia will first have more than \(\$28\,000\) in her bank account during the year \(2039\).[3]
The cost of the boat at the start of \(2026\) was \(\$28\,000\), and it depreciates at a constant annual rate of \(r\%\). After one year the cost of the boat is \(\$26\,222\).
Find the value of \(r\).[2]
Due to this depreciation, Lavinia will be able to buy the boat before \(2039\). She will buy the boat as soon as she has enough money in her bank account to pay for it.
Determine the year during which Lavinia will buy the boat.[5]