For the next 10 years, Debra will receive a $\$ 10,000$ check at the end of every year as part of an insurance settlement. (a) Assuming she were to deposit that money into an annuity at $4.5 \%$ compounded continuously, how much would her annuity be worth at the end of the 10 years? (b) She is offered a one-time lump sum amount payment, the amount being what she would need now that would grow to the future value found in part (a) at the same percentage rate. What is that lump sum? Round your answers to the nearest dollar and do not use commas in the answer blanks. (a) $\$$ (b) $\$$

Step 1 ：Calculate the future value of an annuity with continuous compounding using the formula: \(FV = P \times \frac{e^{rt} - 1}{r}\) where \(P\) is the payment amount, \(r\) is the interest rate, \(t\) is the time in years, and \(e\) is the base of the natural logarithm.

Step 2 ：Substitute the given values into the formula: \(P = 10000\), \(r = 0.045\), \(t = 10\) to calculate the future value.

Step 3 ：Calculate the present value of the future value using the formula: \(PV = FV \times e^{-rt}\) with the same interest rate.

Step 4 ：Substitute the future value calculated in step 2 and the given values \(r = 0.045\), \(t = 10\) to find the lump sum for part (b).

Step 5 ：Round the answers to the nearest dollar for both parts (a) and (b).

Step 6 ：Final Answer for part (a): \(\boxed{126292}\)

Step 7 ：Final Answer for part (b): \(\boxed{80527}\)