It is now December $31,2020($t $=0),$ and a jury just found in favor of a woman who sued the city for injuries sustained in a January $2019$ accident. She requested recovery of lost wages plus $$500,000$ for pain and suffering plus $$100,000$ for legal expenses. Her doctor testified that she has been unable to work since the accident and that she will not be able to work in the future. She is now $62,$ and the jury decided that she would have worked for another $3$ years. She was scheduled to have earned $$36,000$ in $2019.($To simplify this problem, assume that the entire annual salary amount would have been received on December $31,2019.)$ Her employer testified that she probably would have received raises of $3\%$ per year. The actual payment for the jury award will be made on December $31,2021.$ The judge stipulated that all dollar amounts are to be adjusted to a present value basis on December $31,2021,$ using a $7\%$ annual interest rate and using compound, not simple, interest. Furthermore, he stipulated that the pain and suffering and legal expenses should be based on a December $31,2020,$ date. How large a check must the city write on December $31,2021?$ Do not round intermediate calculations. Round your answer to the nearest dollar.

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Your father is $50$ years old and will retire in $10$ years. He expects to live for $25$ years after he retires, until he is $85.$ He wants a fixed retirement income that has the same purchasing power at the time he retires as $$55,000$ has today. $($The real value of his retirement income will decline annually after he retires.$)$ His retirement income will begin the day he retires, $10$ years from today, at which time he will receive $24$ additional annual payments. Annual inflation is expected to be $4\%.$ He currently has $$85,000$ saved, and he expects to earn $7\%$ annually on his savings. How much must he save during each of the next $10$ years $($end$-$of$-$year deposits$)$ to meet his retirement goal? Do not round intermediate calculations. Round your answer to the nearest dollar.

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A father is now planning a savings program to put his daughter through college. She is $13,$ plans to enroll at the university in $5$ years, and she should graduate $4$ years later. Currently, the annual cost $($for everything $-$ food, clothing, tuition, books, transportation, and so forth$)$ is $$13,000,$ but these costs are expected to increase by $6\%$ annually. The college requires total payment at the start of the year. She now has $$9,500$ in a college savings account that pays $8\%$ annually. Her father will make six equal annual deposits into her account; the first deposit today and sixth on the day she starts college. How large must each of the six payments be$?($Hint: Calculate the cost $($inflated at $6\%)$ for each year of college and find the total present value of those costs, discounted at $8\%,$ as of the day she enters college. Then find the compounded value of her initial $$9,500$ on that same day. The difference between the PV of costs and the amount that would be in the savings account must be made up by the father's deposits, so find the six equal payments that will compound to the required amount.$)$ Do not round intermediate calculations. Round your answer to the nearest dollar.

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