Sunrise Industries wishes to accumulate funds to provide a retirement annuity for its vice president of research, Jill Moran. Ms$.$ Moran, by contract, will retire at the end of exactly $12$ years. Upon retirement, she is entitled to receive an annual end$-$of$-$year payment of $$42,000$ for exactly $20$ years. If she dies prior to the end of the $20-$year period, the annual payments will pass to her heirs. During the $12-$year $$accumulation period,$$ Sunrise wishes to fund the annuity by making equal, annual, end$-$of$-$year deposits into an account earning $9\%$ interest. Once the $20-$year $$distribution period$$ begins, Sunrise plans to move the accumulated monies to an account earning a guaranteed $12\%$ per year. At the end of the distribution period, the account balance will equal zero. Note that the first deposit will be made at the end of year $1$ and that first distribution payment will be received at the end of year $11\mathrm{.}3.$

$1.$ How large a sum must Sunrise accumulate by the end of year $12$ to provide the $20-$year, $$42,000$ annuity?

$2.$ How large must Sunrise$$s equal, annual, end$-$of$-$year deposits into the account be over the $12-$year accumulation period to fund fully Ms$.$ Moran$$s retirement annuity?

$3.$ How much would Sunrise have to deposit annually during the accumulation period if it could earn $10\%$ rather than $9\%$ during the accumulation period?

$4.$ How much would Sunrise have to deposit annually during the accumulation period if Ms$.$ Moran$$s retirement annuity were a perpetuity and all other terms were the same as initially described?

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