The one$-$year probability of the failure of a newly purchased machine is given by qx $=$

$($x $+1)/10$

for integer values of x the age of the machine. The CONTINUOUSLY COMPOUNDING interest rate is $\backslash$delta $=8\%.$

$($a$)$ Compute the EPV of a discrete insurance benefit on a new machine $($age x $=0)$ that pays $$30,000$ at the end of the year in which the machine fails.

$($b$)$ USING YOUR ANSWER FROM PART $($a$),$ find the EPV of the same insurance benefit if it pays at the end of the month in which the failure occurs. You will assume failure follows a UDD during each year and use the relation involving the ratio of interest rates.

$($c$)$ USING YOUR ANSWER FROM PART $($a$)$ again, recompute the answer to part $($b$)$ using the other approximation that was determined involving compounding to a fractional power based on the compounding frequency.

Note that failing to use the requested approximation and instead calculating the value of the monthly insurance directly could result in the answer being marked as incorrect.