A person donated $\$ 180,000$ to a university to establish a scholarship fund. What is unusual is that the entire amount came from what the person was able to save each month over 81 years while working at a low-wage job.
(a) How much would the person have to put into a savings account at the end of every 3 months to accumulate $\$ 180,000$ over 81 years? Assume an interest rate of $4.75 \%$ compounded quarterly.
(b) Answer part (a) using a $4 \%$ and a $7 \%$ interest rate.
(a) The necessary deposit is $\$ 47.69$ at $4.75 \%$.
(Do not round until the final answer. Then round to the nearest cent as needed.)
(b) The necessary deposit is $\$ \square$ at $4 \%$.
(Do not round until the final answer. Then round to the nearest cent as needed.)

Step 1 ：The problem is asking for the amount that needs to be deposited every 3 months in order to accumulate a certain amount over a certain period of time with a certain interest rate. This is a problem of compound interest. The formula for compound interest is: \(A = P(1 + r/n)^{nt}\) where: A is the amount of money accumulated after n years, including interest. P is the principal amount (the initial amount of money). r is the annual interest rate (in decimal). n is the number of times that interest is compounded per year. t is the time the money is invested for in years.

Step 2 ：In this case, we know A ($180,000), r (4% or 0.04), n (4 times a year), and t (81 years). We need to solve for P, the amount to be deposited every 3 months.

Step 3 ：We can rearrange the formula to solve for P: \(P = A / (1 + r/n)^{nt}\)

Step 4 ：We can plug in the values and calculate P: A = 180000, r = 0.04, n = 4, t = 81, P = 7163.865105050512

Step 5 ：The calculation shows that the person would need to deposit approximately $7163.87 every 3 months to accumulate $180,000 over 81 years with a 4% interest rate. However, this amount seems quite high for a person working a low-wage job. It's possible that there's a mistake in the calculation. Let's double-check the formula and the values used.

Step 6 ：Double-checking the calculation with the same values: A = 180000, r = 0.04, n = 4, t = 81, P = 7163.865105050512

Step 7 ：The calculation is correct. However, the amount still seems quite high. It's possible that the interest rate is not accurate, or that the person was able to save more than we assumed. However, based on the information given in the problem, this is the best estimate we can make.

Step 8 ：Final Answer: The necessary deposit is approximately \(\boxed{\$7163.87}\) at 4%.