2021 May TZ1 P2 Q7

Two friends Amelia and Bill, each set themselves a target of saving \(\$20 \ 000\). They each have \(\$9000\) to invest.

1. Amelia invests her \(\$9000\) in an account that offers an interest rate of \(7\%\) per annum compounded annually.

   1. Find the value of Amelia's investment after \(5\) years to the nearest hundred dollars.

   2. Determine the number of years required for Amelia's investment to reach the target.

      [5]

2. Bill invests his \(\$9000\) in an account that offers an interest rate of \(r\%\) per annum compounded monthly, where \(r\) is set to two decimal places.

   Find the minimum value of \(r\) needed for Bill to reach the target after \(10\) years.

   [3]

3. A third friend Chris also wants to reach the \(\$20 \ 000\) target. He puts his money in a safe where he does not earn any interest. His system is to add more money to this safe each year. Each year he will add half the amount added in the previous year.

   1. Show that Chris will never reach the target if his initial deposit is \(\$9000\).

   2. Find the amount Chris needs to deposit initially in order to reach the target after \(5\) years. Give your answer to the nearest dollar.

      [8]