what + wathawul Who Murdered Maxwell Murphy? During a cold and rainy March night, the police were summoned to the home of the rich and famous art collector Maxwell Murphy. Someone had murdered Mr. Murphy in his bedroom during the night. During that time, there were four other people in the house besides Mr. Murphy. Those people are now the main suspects: his butler, the museum curator, his son, and his son's girlfriend. After investigating the scene of the crime, and questioning the four suspects, the police determined that Mr. Murphy was last seen alive at the conclusion of dinner around 9:30 pm. His dead body was then discovered at 11:00 pm. The police also noted that after the murderer killed Mr. Murphy, he/she left through the window of his bedroom, shattering the glass in the process. However, the police were still puzzled as to which of the four suspects committed the crime and what approximate time the murder occurred. They decided to call in an expert team of mathematicians to help. Your group, being the expert mathematicians, must now use what you know about logarithmic and exponential equations to solve the mystery of who murdered Maxwell Murphy. a ) It took a few hours for the police to investigate so you do not arrive at the Murphy mansion until 1:15 am. Your first task is to try and estimate the time of death using Newton's Law of Cooling. At 1:30 am, the corpse had a temperature of 79.5 F. According to the thermostat on the wall, the temperature of the room was 72 F and it is safe to assume that the temperature had been at that level for the entire night. Using these values, as well as the normal body temperature of 98.6 F, estimate the time of death. 6 ) However, in the town of Mathsville, there was a strain of the flu that started going around twelve days ago, and it may be possible that Mr. Murphy had it. The town of Mathsville is quite small, with a population of only 5,000 residents. Using the equation for the spread of the flu virus, determine what percent of the town had the flu. 28 1 = yP-510 00 4 ( Th c) If more that 60% of the population of the town had the flu, then it is probable that Mr. Murphy was also suffering from the flu at the time of his death. If so, the temperature of his body at the time of death would have been 100 F. Use this information in the Newton's Law of Cooling equation to find a second possible time of death to give you a time window for when the murder occurred. d) While you are calculating the time of death, the police inform you that Mr. Murphy had just closed a bank account at the local bank in Mathsville. As a result, there was a large sum of money in the house at the time of death. While the police do not know how much was withdrawn when the account was closed, they do know that the account was opened five years ago and the original deposit was two hundred thousand dollars. Since this is the only bank in Mathsville, you also have an account there. You know that the interest rate on an account is 5.75% and the interest is compounded monthly. As a result, you calculate the amount of money that was in the account at the time of its closing. Mr. Murphy still had one hundred thousand dollars in his desk drawer, so it's possible the murdered may have stolen the rest. Determine how much money is missing. Now that you have a time frame for the time of death and a possible motive, it's time to look at the suspects and try to determine which one to charge with the murder of Maxwell Murphy.The Suspects Suspect #1 Name: Henry Scoughings Relation to Mr. Murphy: personal butler and friend Whereabouts: seen throughout the evening alone in the kitchen but could possibly have been free at any time to commit the murder. He discovered the body of Mr. Murphy at 11:00 pm and called police. e) Henry the butler was a very loyal servant to Mr. Murphy for many years. However, one person had started a rumor that due to Henry's advanced age, Mr. Murphy was considering replacing him with a new butler. The rumor had started in the town three weeks ago and if Henry had heard about it, it probably would have caused him to become very upset, possibly even angry enough to murder Mr. Murphy and rob him. However, Henry keeps to himself and would have been one of the last people to hear the rumor. Keeping in mind that there are 5,000 people in Mathsville, determine if there was sufficient time for the entire town to hear about this rumor. f) Even if Henry did not commit the murder out of anger, he could have still done it for the money. The police discovered that Henry was considering retiring in 5 years. However, to live comfortably in retirement, he wanted to make sure he had $200,000 saved up for his retirement. He opened a retirement account 30 years ago when he began working for Mr. Murphy and would deposit $150 into the account each month. The account averaged 6% interest over the lifetime of the annuity. Use the future value of an annuity formula to determine how much the account will be worth 5 years from now to determine is Henry needed to rob Mr. Murphy for money. Finally, from his spot in the kitchen, Henry did not report hearing anything out of the ordinary during the night, despite the fact that the murderer broke a window during his/her exit. Because the kitchen is not near the bedroom, for Henry to hear the glass breaking, the noise would have to be at least 75 decibels. If the intensity of breaking glass is 10-10 watts/cm', use the Intensity of sound equation to determine if Henry could have heard the glass breaking from his spot in the kitchen. Suspect #2 im deposited $90,009 Name: Marvin Murphy Relation to Mr. Murphy: Son and heir to the Murphy fortune buy pointing Whereabouts: the den the entire night following dinner. Tell him cost $80,900 0 h ) Marvin never really liked his dad's paintings. He wanted to sell them and use the money to bankroll his career as a jazz musician. He needed $50,000 to get a good start on his career. His dad had originally said that he would eventually sell one of his paintings and give Marvin the money from the sale. However, Mr. Murphy had recently changed his mind about selling the painting. This had angered Marvin at the time. The painting that was being considered was bought 5 years ago for $25,000 and has an appreciation rate of 10% per year. Since he did not know of his father recently closed account, is the painting presently worth enough for Marvin to murder his dad and sell the painting to start his career? i) Marvin spent the evening with his girlfriend in the den listening to jazz music. He left the den briefly at 9:50 pm to escort his girlfriend to her car and then returned to the den where he was when the police arrived. He claimed he did not hear anything out of the ordinary. For him to hear the sound of the glass shattering over his jazz music, the sound would have to be 90 decibels. Determine if Marvin could have heard the glass breaking from the den.Suspect #3 Name: Bonnie Ballyhoo Relation to Mr. Murphy: longtime girlfriend of Mr. Murphy's son Whereabouts: in the den with Marvin until 9:50 when she left to go home, where she was from 10:00 until the time police contacted her 1 ) Bonnie's father, Bert Ballyhoo, was a very famous painter in Mathsville. Mr. Murphy bought the original of one of his paintings. After her father's death at a very young age, Bonnie had repeatedly said she wanted the painting back in the Ballyhoo family. She also had wanted to purchase several other of her father's paintings from Mr. Murphy and other collectors. Her father opened an account for Bonnie 24 years ago when she was born to help her throughout her life. The bank account can only have a maximum of 30% of its total value withdrawn in a year. The account gets compounded monthly, but gets a special interest rate of 6%. The initial deposit in the account was $50,000. She knows the Maxwell family will sell her the paintings she wants for $100,000. Determine if she has enough money to withdraw from here account to buy the paintings. If not, it is possible she knew of the closed account and murdered Mr. Murphy to get the money she needed. k) While Bonnie was in the Murphy mansion, she spent the evening with Marvin in the den listening to jazz music. However, she left briefly to look at her father's painting in the gallery at around 9:45 pm. She then returned to the den and Marvin escorted her to her car at 9:50 pm. She did not hear anything out of the ordinary while she was alone in the gallery. For her to have heard the broken glass, it would have had to be at least 60 decibels. Determine if Bonnie should have heard the breaking glass from the gallery. Suspect #4 Name: Joseph Hutchinson Relation to Mr. Murphy: curator of the art museum housing some of Mr. Murphy's paintings Whereabouts: Was at the Murphy mansion between 10:00 pm and 10:30 pm 1 ) Joseph also would like to buy some of Mr. Murphy's paintings to ensure they stay at the museum. 10 years ago, the museum deposited $50,000 into an account in a bank outside of Mathsville where the interest is compounded continuously with a 6.75% interest rate. At the end of last year the money was withdrawn and Joseph has the ability to use it to buy paintings and artifacts for the museum with the money. The paintings Mr. Murphy was planning to sell him cost $80,000. Did Joseph have enough to buy them all? If not he may have murdered Mr. Murphy to rob him and then buy the paintings from Marvin. Joseph also spent some time the night of the murder looking at a manuscript that Mr. Murphy m ) had recently purchased. He was supposed to do a carbon dating of it to determine how old the manuscript was. If it was over 2000 years old, then it would be extremely valuable. If this is the case, Joseph could have murdered Mr. Murphy and buy the manuscript from Marvin, since Marvin would not know of its true value. Joseph had arrived at the mansion at 10:00 to meet with Mr. Murphy about the manuscript. However, Mr. Murphy was unable to see him at that time and Joseph left the mansion at around 10:30 pm. Joseph was going to tell Mr. Murphy that the manuscript contained 78% as much carbon 14 as a new piece of paper of the same makeup. According to him then, the manuscript was not nearly as old as Mr. Murphy originally thought. Using the half-life formula, the radioactive decay formula, and knowing that the half-life of carbon 14 is 5730 years, determine if Joseph is being truthful.While Joseph was at the Murphy mansion, he spent some time in the gallery looking at Mr. Murphy's paintings. While in the gallery he said he heard no strange noises. For him to hear the glass breaking, it would have had to be at least 60 decibels. Determine if Joseph should have heard the glass breaking. Create a report that includes all the details about the murder and all the suspects' claims. Determine who police should charge with the murder of Maxwell Murphy. Be sure that you back up your report with sounds reasoning and plenty of mathematics! The characters in this story are fictitious and any similarities to any person living or deceased are purely coincidence.Newton's Law of Cooling Intensity of Sound T = A+(B- A)e-OAl B(1) = 1010g 3 10 - 16 T = Current body temperature A = Ambient temperature of the room B(1) = level of sound in decibels (temperature of room) B = Body temperature at time of death I = intensity of sound t = hours since death Half-Life Formula Spread of a Flu Virus k= 1+(P-1)e -0.81 number of people infected k = decay rate t = half life in years P = population t = time in days Exponential Growth/Decay Model Compound Interest A = Neki A = P ( 1 + * ) " A = Final amount N = original amount k = growth/decay rate P = initial investment t = time in years A= final amount r = interest rate Continuously Compounded Interest t = time in years n = number of times compounded in a A = Pe" year A = Final amount Spread of a Rumor P = original deposit r = interest rate P t = time in years H = - 1 + (P-S)e-041 Future Value of an Annuity H = number of people who have heard the rumor F = P (1 + i ) " P = population i S= number of people who start the Fn = Final value rumor P = Payment t = time in days i = interest rate/payments per year n = payments per year x num. of years