Question: Mara and Braylen are both age $75$. When Mara was $25$ years old, she began depositing $1200$ per year into a savings account. She made deposits for the first $10$ years, at which point she was forced to stop making deposits. However, she left her money in the account, where it continued to earn interest for the next $40$ years. Braylen didn't start saving until he was $47$ years old, but for the next $28$ years he made annual deposits of $1200$. Assume that both accounts earned an average annual return of $6\%$ (compounded once a year). Complete parts (a) through (d) below. Part 1 a. How much money does Mara have in her account at age $75$?

1. **State the problem:** Calculate the amount of money Mara has in her account at age $75$ given she deposited $1200$ per year for $10$ years starting at age $25$, then left the money to grow for $40$ more years at an annual interest rate of $6\%$ compounded yearly. 2. **Formula used:** We use the future value of an annuity formula for the first $10$ years of deposits: $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where: - $P = 1200$ (annual deposit) - $r = 0.06$ (annual interest rate) - $n = 10$ (number of deposits) After the deposits stop, the amount grows for $40$ years with compound interest: $$FV_{total} = FV \times (1 + r)^m$$ where $m = 40$ years. 3. **Calculate the future value of the annuity (deposits):** $$FV = 1200 \times \frac{(1 + 0.06)^{10} - 1}{0.06}$$ Calculate $(1 + 0.06)^{10}$: $$1.06^{10} = 1.790847$$ So, $$FV = 1200 \times \frac{1.790847 - 1}{0.06} = 1200 \times \frac{0.790847}{0.06}$$ Simplify the fraction: $$\frac{0.790847}{0.06} = 13.18078$$ Therefore, $$FV = 1200 \times 13.18078 = 15816.94$$ 4. **Calculate the amount after $40$ years of growth without deposits:** $$FV_{total} = 15816.94 \times (1.06)^{40}$$ Calculate $(1.06)^{40}$: $$1.06^{40} = 10.2857$$ Multiply: $$FV_{total} = 15816.94 \times 10.2857 = 162592.21$$ 5. **Final answer:** At age $75$, Mara has $162592.21$ in her account. This matches the given answer, confirming the calculations.