Police arrive at a murder scene at 3 : 30 AM and immediately record the​ body's temperature which was 90degrees°F. At 5:30 ​AM, after thoroughly inspecting and fingerprinting the​ area, they again took the temperature of the body which had dropped to 85degrees°F. The temperature of the crime scene has remained at a constant 70degrees°F. Determine when the person was murdered.​(Assume that the victim was healthy at the time of death. That​ is, assume that the temperature of the body at the time of death was 98.6degrees°​F.) The person was murdered at ? . ​(Do not round until the final answer. Then round to the nearest minute as​needed.)

Answer:The person was murdered at 11.40 PMStep-by-step explanation:Given information :Initial body temperature, [tex]T_{0}[/tex] = 90[tex]^{o}F[/tex] Police arrive at a murder scene at 3 : 30 AM, at  5.30 AM (120 minutes), temperature of the body had dropped to 85 [tex]^{o}F[/tex].T(120) = 85 [tex]^{o}F[/tex] The temperature of the crime scene, C = 70[tex]^{o}F[/tex] to determine the time of the murderer, we can use the Newton's lau of cooling:T(t) = C + ([tex]T_{0}[/tex] - C) [tex]e^{-kt}[/tex]whereT(t) = temperature at any given time C = surrounding temperature [tex]T_{0}[/tex] = initial temperature of heated object k = cooling constantthusT(t) = C + ([tex]T_{0}[/tex] - C) [tex]e^{-kt}[/tex]T(120) = 70 + (90 - 70) [tex]e^{-120k}[/tex]85 = 70 + 20 [tex]e^{-120k}[/tex][tex]e^{-120k}[/tex] = (85-20) / 70-120 k = ln (65/70)k = - ln (65/70) / 120   = 0.00062so we have the cooling equationT(t) = 70 + ([tex]T_{0}[/tex] - 70) [tex]e^{-0.0062t}[/tex]The temperature at the time of the death is 98.6[tex]^{o}F[/tex]  which was the initial temperature of the person murdered, the temperature dropped to T(t) =  85 [tex]^{o}F[/tex] , now we can find the time of him being murdered.T(t) = 70 + (98.6 - 70) [tex]e^{-0.0062t}[/tex]85 = 70 + 28.6 [tex]e^{-0.0062t}[/tex][tex]e^{-0.0062t}[/tex] = (85- 28.6)/70-0.00062t = ln ((85- 28.6)/70)t = - ln ((85- 28.6)/70)/ 0.00062t = 350 minutes therefore, the preson being murdered 350 minutes (5 hours 50 minutes) before 5.30, which was 11. 40 PM