$(20pts)$ Joe wants $to$ purchase a perpetuity paying $50$ per year with the first payment due $at$ the end

$of$ year $9.He$ can purchase $itby$ either:

$(a)$ paying $40$ per year $at$ the end $of$ each year for $8$ years; $or$

$(b)$ paying $X$ per year $at$ the end $of$ each year for the first $4$ years and nothing for the next $4$ years.

All the annuities are based $on$ annual effective interest rate $ofi,i>0.$

Calculate $X$