$($Related to Checkpoint $5\mathrm{.}6)($Solving for i$)$ You are considering investing in a security that will pay you $$4,000$ in $31$ years.

a$.$ If the appropriate discount rate is $10$ percent, what is the present value of this investment?

b$.$ Assume these investments sell for $$1,684$ in return for which you receive $$4,000$ in $31$ years. What is the rate of return investors earn on this investment if they buy it for

$$1,684?$

a$.$ If the appropriate discount rate is $10$ percent, the present value of this investment is $

$($Round to the nearest cent.$)$