A woman deposits $\$ 13,000$ at the end of each year for 6 years in an account paying $4 \%$ interest compounded annually.
(a) Find the final amount she will have on deposit.
(b) Her brother-in-law works in a bank that pays $3 \%$ compounded annually. If she deposits money in this bank instead of the other one, how much will she have in
(c) How much would she lose over 6 years by using her brother-in-law's bank?
(a) She will have a total of $\$ 86228.68$ on deposit.
(Simplify your answer. Round to the nearest cent as needed.)
(b) She will have a total of $\$ \square$ on deposit in her brother-in-law's bank.
(Simplify your answer. Round to the nearest cent as needed.)

Step 1 ：First, we need to understand the problem. The woman deposits $13,000 at the end of each year for 6 years in an account paying $4 \%$ interest compounded annually. We need to find out how much she will have in her brother-in-law's bank, which pays $3 \%$ compounded annually.

Step 2 ：Let's denote the interest rate in her brother-in-law's bank as $r$, which is $3 \%$ or $0.03$. The total amount she will have on deposit can be calculated by the formula: $$13000(1+r)^5 + 13000(1+r)^4 + 13000(1+r)^3 + 13000(1+r)^2 + 13000(1+r) + 13000.$$

Step 3 ：Substitute $r$ with $0.03$ into the formula, we get: $$13000(1+0.03)^5 + 13000(1+0.03)^4 + 13000(1+0.03)^3 + 13000(1+0.03)^2 + 13000(1+0.03) + 13000.$$

Step 4 ：Calculate the expression step by step, we get: $$13000(1.03)^5 + 13000(1.03)^4 + 13000(1.03)^3 + 13000(1.03)^2 + 13000(1.03) + 13000 = 13000(1.15927407) + 13000(1.12550881) + 13000(1.09262701) + 13000(1.06050902) + 13000(1.03) + 13000.$$

Step 5 ：Continue to calculate, we get: $$15020.56 + 14616.11 + 14204.15 + 13786.62 + 13390 + 13000 = \boxed{\$ 84017.44}.$$

Step 6 ：Check the result, the total amount she will have on deposit in her brother-in-law's bank is $\$ 84017.44$, which meets the requirements of the problem.