$(1)$ A continuous annuity is a steady stream of money that is paid to a person or group out of an interest bearing account. Such an annuity may be established, for exam$-$ ple, by making as initial deposit in a savings account and then making continuous withdrawals to pay the annuity. Suppose an initial deposit of $$5,400,000$ is put into an account which earns $5\mathrm{.}5\%$ interest compounded continuously. Also suppose you immediately start withdrawals at a rate of $$300,000$ per year. $($a$)$ Set up a differential equation that is satisfied by y$($t$),$ the amount of money in the account at time t$,$ measured in years. $($b$)$ Sketch the solution to this equation given by the initial condition provided. $($c$)$ What happens to the account in the long run? $($d$)$ If you wanted to live off the $$300,000$ per year for the rest of your life, how much money do you need to initially deposit? $($Note: This is a nice way to live, if you can swing it$!)$