$($Related to Checkpoint$5\mathrm{.}6)($Solving for i$)$ Springfield Learning sold zero coupon bonds$($bonds that$$ don't pay any$$ interest, instead the bondholder gets just one$$ payment, coming when the bond$$ matures, from the$$ issuer$)$ and received $ $1,000$ for each bond that will pay $$$$19,000$ when it matures in $35$ years.

a$.$At what rate is Springfield Learning borrowing the money from$$ investors?

b$.$If Nancy Muntz purchased a bond at the offering for $$$$1,000$ and sold it $20$ years later for the market price of $ $3,100,$ what annual rate of return did she$$ earn?

c$.$If Barney Gumble purchased$$ Muntz's bond at the market price $($$$3,100)$ and held it $15$ years until$$ maturity, what annual rate of return would he have$$ earned?