$($a$)$ What rate of interest with continuous compounding is equivalent to an $7\%$

interest rate with semi$-$annual compounding?

$($b$)$ Consider a project costing $30,000$ for each of the first three years $($where the

first payment is made today at time $T=0).$ Then, starting at the beginning

of the fourth year $($i$.$e$.$ at time $T=3),$ it begins generating a profit of $18,000$

per year for all future years. Assume an annually compounded interest rate of

$4\%.$

$($i$)$ Write down the present value of this project at the beginning of year $3+t,$

where $t$ is any non$-$negative integer.

$($ii$)$ Find the value of $T$ at which you break even.