Christine$$ O'Brien, who is$$ self$-$employed, wants to invest $$$$40$ comma $000$ in a pension plan. One investment offers $7\%$ compounded quarterly. Another offers $6\mathrm{.}75\%$ compounded continuously.

a$.$ Which investment will earn the most interest in $7$years$?$

b$.$ How much more will the better plan$$ earn?

c$.$ What is the effective rate in each$$ case?

d$.$ If Ms$.$ O'Brien chooses the plan with continuous$$ compounding, how long will it take for her $$$$40$ comma $000$ to grow to $$$$70$ comma $000?$

e$.$ How long will it take for her $$$$40$ comma $000$ to grow to at least $$$$70$ comma $000$ if she chooses the plan with quarterly$$ compounding? $($Be$$ careful; interest is added to the account only every$$ quarter.$)$