Monthly compound interest savings compared with a depreciating price to find the purchase year

A music school wants to buy a grand piano priced at \(\$40\,000\). On \(1\) January \(2027\) it places \(\$26\,000\) in a savings account paying a nominal annual interest rate of \(4.2\%\), compounded monthly.

Interest is added to the account on the last day of every month. The school makes no other payments into or out of the account.

1. Find the amount in the account on \(1\) January \(2032\), correct to the nearest dollar.[3]
2. Show that the amount in the account first exceeds \(\$40\,000\) during \(2037\).[3]
3. On \(1\) January \(2027\) the piano is priced at \(\$40\,000\). Its price falls at a constant rate of \(r\%\) per year, and one year later it is priced at \(\$38\,060\).
   Find the value of \(r\).[2]
4. Because the price is falling, the school can afford the piano before \(2037\). It buys the piano at the first moment the account holds at least the current price.
   Determine the year in which the school buys the piano.[5]