State-sponsored lotteries are extremely popular and highly successful methods by which state governments in many countries raise much-needed funds for financing public expenses, especially education. In Michigan alone, during the year 2015, Michigan Lottery reported annual sales of $2.7 billion, and generated $795.5 million in net revenue for the state School Aid Fund, supporting public education (K-12) programs through- out the state. Retailers received annual commissions of $203.6 million, while Michigan Lottery players collected prizes worth $1.69 billion. Table 1 presents sales and funding figures accounted for by the Michigan Lottery since its inception in 1972 up until fiscal year 2015. The numbers are quite impressive! Mega Millions is a multi-state draw game, formerly known as The Big Game. On January 31, 2010, the Multi-State Lottery Association (MUSL), an association of 37 member lotteries, joined with the 10 participating Mega Millions states including California, Georgia, Illinois, Massachusetts, New Jersey, New York, Ohio, Virginia, and Washington to participate in the Mega Millions game with Michigan. On December 12, 2010, Megaplier, an add-on game to Mega Millions, began where players can increase their prize by two to five times. The Megaplier number is selected at random before each drawing. If a player wins a non-jackpot prize on their Mega Millions wager, their prize is increased by the Megaplier number drawn and they can win up to $5 million. Beginning October 19, 2013, the Mega Millions prize matrix changed: players select from two pools of numbers, five white ball with numbers between 1 and 75, and one gold mega balls number between 1 and 15 for a chance to win a multimillion-dollar jackpot prize. The jackpot starts at $15 million and is guaranteed to increase $5 million with every time it rolls. This game offers a second prize of $1 million and nine ways to win with guaranteed prize amounts at lower prize levels. Drawings are conducted twice a week on Tuesday and Friday. Powerball is another multi-state draw game in which Michigan Lottery participates. It is considered nationally to be the “marquee big jackpot game.” It was introduced on January 31, 2010. Players select 5 of 59 white balls and 1 of 39 red balls for a chance to win a multimillion-dollar jackpot. On January 15, 2012, the Powerball game was reintroduced at a $2 price point, with jackpots starting at $40 million. This game offers nine ways to win, with guaranteed prize amounts on lower prize levels, and includes a $1 million second prize. Drawings are conducted twice a week on Wednesday and Saturday. Power Play is an add-on game to Powerball, with a fixed prize-structure that includes prizes ranging from $8 to $2 million. A total of 47 member lotteries, Mega Millions and MUSL members participate in the Powerball game with Michigan. In January 2016, the record-shattering Powerball jackpot amounting to $1.5864 billion was won by three ticket holders, John and Lisa Robinson, Maureen Smith, and Mae and Marvin Acosta. Each winner was awarded a nominal sum of $528.8 million. The Michigan Lottery can pay Powerball and Mega Millions jackpot winnings in one of two ways: as an annuity or in one lump-sum/cash-option payment for the present cash value of the jackpot share. When a winner selects annuity payments, the jackpot is paid out in equal installments over 30 years. When a winner selects the cash option, the Lottery pays the winner the present cash value of the announced jackpot in one lump-sum payment. In effect, the Lottery takes all of the money that would have been invested to fund the 30-year annuity and turns it all over to the winner, retaining absolutely none of the prize. All three winners of the January 2016 Powerball jackpot opted to take a cash-option payment amounting to $327.8 million per ticket. Regardless of which option the winner selects, the Michigan Lottery is required by law to withhold estimated income taxes for federal (25%) and state (4.9%) on any prize over $5,000. These amounts are estimates only, and the winner is required to satisfy any further tax liability for the year in which the prize award is claimed. Table 1 Totals: 1972 Start-up through FY 2015* Total Lottery Ticket Sales $57.84 billion Net Revenue to Aid Education $19.57 billion Retailer Commissions $ 3.77 billion Prizes to Players $31.58 billion *http://www.michiganlottery.com/about_us?#financial Questions: If you were one of the three winners of the 2016 Powerball jackpot, which option would you select? Why? If you decide to select the annuity option, how much money would you receive each year after taxes? What discount rate is the Powerball lottery using when deciding to pay a lump sum of $327.8 million to each of the three winning ticket holders? Is the State of Michigan justified in advertising the prize amount as $1.586 billion? Explain. If the only option available was an annuity payment plan, what could the winner do to maximize the value of his/her winnings assuming that the risk-free rate of interest is 5%? Why do most winners select the cash option plan when given a choice? If Michigan Lottery would like to give the annuity option an equal chance of being selected, how would it have to structure its payments?
> A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?
> The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm2, find the dimensions of the poster with the smallest area.
> A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 1.1.62.) If the perimeter of the window is 30 ft, find the dimensions of the window so that
> A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible surface area of such a cylinder.
> A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.
> A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder.
> If the two equal sides of an isosceles triangle have length a, find the length of the third side that maximizes the area of the triangle.
> Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r.
> Find the area of the largest trapezoid that can be inscribed in a circle of radius 1 and whose base is a diameter of the circle.
> Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.
> Differentiate the function. g(x) = ln(xe-2x)
> Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.
> Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.
> Find, correct to two decimal places, the coordinates of the point on the curve y = sin x that is closest to the point (4, 2).
> Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).
> Find the point on the line y = 2x + 3 that is closest to the origin.
> (a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.
> If the farmer in Exercise 18 wants to enclose 8000 square feet of land, what dimensions will minimize the cost of the fence? Exercise 18: A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed
> A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fe
> Do Exercise 16 assuming the container has a lid that is made from the same material as the sides. Exercise 16: A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the bas
> A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for t
> If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
> A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.
> A farmer wants to fence in an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?
> Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a)
> Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw s
> Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as possible.
> Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.
> Find expressions for the first five derivatives of f (x) = x2ex. Do you see a pattern in these expressions? Guess a formula for f(n) (x) and prove it using mathematical induction.
> What is the minimum vertical distance between the parabolas y = x2 + 1 and y = x - x2?
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f(x) = 6 sin x
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f (x) = x6 - 5
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f (x) = 22x6 +
> Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ’ and f ’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. f (x) = x5 - 5
> (a) Investigate the family of polynomials given by the equation f (x) = 2x3 + cx2 - 2x. For what values of c does the curve have maximum and minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the curve y = x
> (a) Investigate the family of polynomials given by the equation f (x) = cx4 - 2x2 + 1. For what values of c does the curve have minimum points? (b) Show that the minimum and maximum points of every curve in the family lie on the parabola y = 1 - x2. Illu
> Investigate the family of curves given by the equation f (x) = x4 + cx2 + x. Start by determining the transitional value of c at which the number of inflection points changes. Then graph several members of the family to see what shapes are possible. Ther
> Differentiate the function. f(x) = log 10 (1 + cos x)
> Investigate the family of curves given by f (x) = xe-cx, where c is a real number. Start by computing the limits as x ( ±∞. Identify any transitional values of c where the basic shape changes. What happens to the maximum or minimum points and inflection
> The family of functions f (t) = C(e-at – e-bt), where a, b, and C are positive numbers and b . a, has been used to model the concentration of a drug injected into the bloodstream at time t = 0. Graph several members of this family. What do they have in c
> Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should als
> Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should als
> Find dy/dx by implicit differentiation. x2 - 4xy + y2 = 4
> In Example 4 we considered a member of the family of functions f (x) = sin(x + sin cx) that occur in FM synthesis. Here we investigate the function with c = 3. Start by graphing f in the viewing rectangle [0, π] by [-1.2, 1.2]. How many local maximum po
> A company operates 16 oil wells in a designated area. Each pump, on average, extracts 240 barrels of oil daily. The company can add more wells but every added well reduces the average daily ouput of each of the wells by 8 barrels. How many wells should t
> Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.
> When John Sullivan was hired as chief investment strategist at the New York headquarters of A. M. Smith Inc., he had indicated that one of his main goals would be to significantly expand the fixed-income unit of the firm’s overall investment portfolio. A
> When Jacqueline and Keith Sommers were “house hunting” five years ago, the mortgage rates were pretty high. The fixed rate on a 30-year mort- gage was 7.25%, while the 15-year fixed rate was at 6.25%. After walking through many homes, they finally reache
> “Boy, this is all so confusing,” said Jason as he stared at the papers on his desk. “If only I had taken the advice of my finance instructor, I would not be in such a predicament today.” Jason Welch, aged 27, graduated five years ago with a degree in foo
> “Greg, the board of directors’ meeting is scheduled two weeks from today, and I’m depending on you to come up with a realistic and honest appraisal of our company’s position,” said Warren, to his assistant Greg Chapman. “I’m sure that there’s more to us
> “Numbers! I need to see numbers!” exclaimed Marcus in response to com- ments made by the assistant vice-president of Finance, Jeff Smith. Marcus Lenovo, president and chief executive officer of Duralex Inc., had been instrumental in significantly increas
> Andy Gillian, the owner of Gillian Pool & Spa Supplies, paced up and down the balcony of his luxurious Victorian home, overlooking a beautiful backyard, which housed a full-size pool/spa and a sprawling, luscious, green lawn. What was worrying Andy w
> The Ultra Cable Corporation, headquartered in Chicago, Illinois, had thus far enjoyed a fairly steady run-up in revenues and profits. Two years ago, it hired Ron Swenson away from the competition to assist the president, Tom Gray, in navigating the compa
> “It’s amazing how much difference there is in the way proposals are presented at two different firms,” said Art Monk to his assistant, Russell Jacobs, as he pointed to the stack of capital investment proposals piled on his desk. “We sure have our work cu
> Howard Cook could hardly wait to get back to his corporate headquarters in Philadelphia, Pennsylvania, and share all the useful information that he had gathered on performance evaluation with his colleagues in the finance department. Howard was vice pres
> As Cecil shuffled through the stack of files on his desk and clicked away on his mouse, his mind kept racing back to what Jason, his boss, had said to him at the last budget meeting. “We can only fund two or three new projects over the next year,” he sai
> Richard Ponting could feel the pressure as he walked into the executive boardroom with his briefcase containing the data and slides pertaining to his latest proposal. The last couple of years had not been very good for him. Two of the projects that he ha
> When Jenny opened her full-service salon and day spa three years ago, she knew that she would have to make some difficult choices regarding the hiring and firing of qualified professionals such as cosmetologists, estheticians, nail technicians, and massa
> If there was one thing the folks at Creative Products Corporation (CPC) knew well, it was how to come up with useful and unique products in the midst of economic adversity. With current year revenues considerably lower and profit margins shrinking due
> When Wayne concocted his cleaning compound some 20 years ago, all that his wife, Corrine, and he were trying to do was to come up with a sweeter, gentler yet tougher, cleaning product. Little did he realize that someday he would be the proud owner of a m
> Alex Peterson, president of Peterson Investments Inc., was faced with a major decision. One of the firms that his company had invested in, Durawear Corporation, was under severe financial distress. It had therefore sent out a proposal for reorganization,
> It was late Sunday night, and Jassir Amor was getting weary. The big presentation was set for 8 am the next day, and Jassir kept remembering what Greg LeBlanc, the chairman of the mergers and acquisitions (M&A) committee had said to him: “The board membe
> It was a hot, humid afternoon in April. Kirk could feel the pressure mounting. The memo on his desk read, “Please see me immediately!” Kirk knew that, sooner or later, his boss, Brian Daltrey, was going to ask him to implement some quick remedies to impr
> “We better get started on this report,” said Hal to his assistant, Roger, as he straightened up in his seat and fired up his laptop computer. “I’m sure Henry will expect a detailed analysis first thing Monday morning.” Hal Burton, vice president for Inte
> “What do you mean we’ve used up all our cash and lines of credit? I don’t get it! I thought we had a healthy financial position as per last year’s financial statements. How could this have happened, Patrick?” said Donald Hunt to his chief accountant. “If
> It was the end of the fourth quarter. The financial statements had been prepared and circulated to the directors of NuSkin Products Inc. (Tables 2 and 3). The firm’s revenues had surpassed the previous quarter’s revenues by over 20%, and the annual sales
> What is the equivalent of native advertising in commercially released movies?
> Why do you think officers and directors of accused companies accept dissent decrees rather than contest accusations at trial?
> Is thinking about committing a crime illegal? The law underlying the case against Microsoft is the Electronic Communications Privacy Act, which was enacted three years before the invention of the World Wide Web. Should that law still apply today? Why or
> Should a manager for an LLC respond to employee complaints of discrimination any differently from a manager at a corporation, a partnership, or a sole proprietorship? Why or why not?
> Consider a company whose employees include both smokers and nonsmokers. The smokers take numerous paid smoking breaks, while the nonsmokers do not. Is there an ethical issue here? Discuss.
> Why might it be fair for the court to hold Coxeter liable for his partner’s fraud?
> Can you think of a way a company can obtain information via an applicant’s social media posts without running the risk of being accused of hiring discrimination?
> How can a company, whether an LLC or some other business form, reduce the chances of discrimination lawsuits?
> How might a couple who enjoy purchasing virtual and digital goods together avoid property division issues in the event of a divorce?
> To what extent must attorneys reveal to their clients where confidential data are stored?
> Radiah Givens was involved romantically with Joseph Rosenzweig. She moved into an apartment on which he made the down payment. She signed the mortgage, but he made the payments and paid household expenses. They later married. She had their marriage annul
> Which companies in Europe do you think may have pressured the European Union to lodge its antitrust complaint against Google?
> At what point might the predicted benefits of a new private commercial endeavor outweigh the constitutional requirement of a taking only for public use?
> How do online escrow services reduce Internet fraud?
> Is it fair that property or wage garnishments may “surprise” a judgment debtor?
> 1. After the initial sequencing change, West Bank provided its customers with a document titled “Miscellaneous Fees” in which a footnote stated, “Checks written on your account will be paid in order daily with the largest check paid first and the smalles
> Does having a digital wallet in an iPhone, Android-based phone, or other smartphone entail more security risks than carrying a physical wallet? Explain.
> Imported beer is not the only product whose labeling may be misleading. For instance, although BMW is a German brand, most BMW X3s and X5s purchased in the United States are actually manufactured in South Carolina. Are there any legal or ethical issues i
> Why does the Bankruptcy Code provide that student loans should not be dischargeable unless there is undue hardship? What argument can be made in favor of allowing student loans to be dischargeable?
> Under what circumstances would you, as a business owner, choose to aggressively defend your business against a customer’s liability lawsuit?
> Some argue that if online retailers are required to collect and pay sales taxes in jurisdictions in which they have no physical presence, they have no democratic way to fight high taxes in those places. Is this an instance of taxation without representat
> To download a specific application (app) to your smartphone or tablet device, usually you have to check a box indicating that you agree to the company’s terms and conditions. Most individuals do so without ever reading those terms and conditions. Print o
> Within Instagram’s current terms of service there is a statement “We may not always identify paid services, sponsored content, or commercial communications as such.” Is it ethical for Instagram to post advertisements without identifying them as advertise
> Why might those entering into contracts be worse off in the long run if the courts increasingly accept impossibility of performance as a defense?
> What might happen if arbitration clauses were prohibited in all consumer contracts?
> If you are running a business, why would you opt to include overly broad waivers in your contracts with consumers?
> How does a manager determine the optimal amount of legal research to undertake to prevent her or his company from violating the many thousands of federal regulations?
> Why might a corporation’s managers agree to pay a large fine rather than to be indicted and proceed to trial?
> Does a manufacturer have to create safety warnings for every product? Why or why not?
> Some companies use internal social media networks for work-related employee communications. Would the same legal rules that apply to monitoring public social media platforms, such as Twitter and Facebook, also apply to company-provided social media platf
> To protect themselves, manufacturers have been forced to include lengthy safety warnings for their products. What is the downside of such behavior?
> Must business owners in all states have to provide the same benefits to employees in a same-sex union as they do to heterosexual couples?
> Review the basic requirements for a valid contract listed at the beginning of this chapter. Now consider the relationship created when a student enrolls in a college or university. (a) One group should analyze and discuss whether a contract has been form
> Should domain name hosting companies be liable for revenge porn?
