Christine O'Brien, who is self$-$employed, wants to invest $$80,000$ 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 $6$ 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 $$80,000$ to grow to $$90,000?$ e$.$ How long will it take for her $$80,000$ to grow to at least $$90,000$ if she chooses the plan with quarterly compounding? $$