Can someone check this work for me? It\'s the project for 7.3 on page P19. Sorry
ID: 3082605 • Letter: C
Question
Can someone check this work for me? It's the project for 7.3 on page P19. Sorry if some of it is hard to read. I could try posting screenshots instead. 1. Solve equation (1), which models the senario in which Joe Wood is killed in the refrigerator. use this solution to esimate the time of death (Recall that normal living body temperature is 98.6 degrees Fahrenheit). Equation (1): dT -- = k(T - Tm) dt Tm= 50 dT -- = k(T - 50) dt I took equation 1 and inserted in the known Tm value which was 50. dT ------ = k dt T - 50 I used seperation of variables to get the equation into a form I could integrate. Integrating results in the equation ln(T-50)= kt+c Taking the e of both sides to get rid of the Ln turn it into T-50=c > exp(kt); Finally just move the 50 to the other side to get print(??); # input placeholder T = 50 + c exp(kt) Using T(0)= 85 to solve for c gets 85 = 50 + c exp(0) subtract 50 from both sides to get 35=c The only other known value is T(-1/2)=84 so using this to solve for k / /-1\ 84 = 50 + 35 exp|k|--|| // 34 / 1 -- = exp|- - k| 35 2 / Takeing the ln of both sides and multiplying by -2 finds k /34 -2 ln|--| = k / k= / /35\ |2 ln|--|| ≈ (0.058) // This matches the statement made by Daphne that k would be a positive value. so my end solution is / /35 T = 50 + 35 exp|2 ln|--| t| / / To find the time of death I set T=98.6 and solve for t. > solve(50+35*exp(2*ln(35/34)*t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 .662355357*60= 40 6-5= 1:00 a.m. - 40 min= 12:20 a.m. if t= 5.66 then the time is about 12:20 a.m. 2. Solve the differential equation (3) using Laplace trenforms. Your solution T(t) will depend on both t and h. (Use the value of k found in Problem 1.) Equation (3): dT -- = k(T - Tm(t)) dt Tm(t)= 50+20* Heaviside(t - h) Subbing Tm(t) into equation 3 gets dT -- = k(T - 50 - 20 Heaviside(t - h)) dt I divide both sides by K before taking the Laplase to get 1 - k *?{dT/dt}=?{T}-?{50}-?{ 20 Heaviside(t - h) } Solving all the Laplase gets sY(s) - T(0) 50 20 exp(-hs) ------------ = Y(s) - -- - ----------- k s s T(0) still equals 85 so subbing that in and moving values around /s 85 50 20 exp(-hs) Y(s)|- - 1| = -- - -- - ----------- k / k s s Next I pull out a 1/k from both sides > ((Y(s))(1/k))(s-k) = (1/k)(85-50*k/s-20*k*exp(-hs)/s); print(??); # input placeholder The 1/k cancles leaving me with 50 k 20 k exp(-hs) Y(s)(s - k) = 85 - ---- - ------------- s s Then I divide by (s-k) to get Y(s)= > 85/(s-k)-50*k/s(s-k)-20*k*exp(-hs)/s(s-k); Next I use partial fractions to break up 1 -------- s(s - k) "1/(s(s-k)=)A/(s)-B/(s-k)" 1=A(s-k)-Bs 1= (A-B)s-Ak A= 1 - - k B= 1 - k so 1 1 1 -------- = - --- + -------- s(s - k) k s k(s - k) > Y(s) = 85/(s-k)-50*k(-1/(k*s)+1/k(s-k))-20*k*exp(-hs)*(-1/(k*s)+1/k(s-k)); / 85 50 50 20 exp(-hs) 20 exp(-hs) Y(s) = +|-----, -- - -----| + ----------- - ----------- s - k s s - k/ s s - k +(85, -50) 50 20 exp(-hs) 20 exp(-hs) Y(s) = ---------- + -- + ----------- - ----------- s - k s s s - k Finally I take a Inverse Laplase to get y(t) y(t)= > {35/(s-k)}/`ℒ`+{50/s}/`ℒ`+{20*exp(-hs)/s}/`ℒ`-{20*exp(-hs)/(s-k)}/`ℒ`; Solving this and subbing /35 2 ln|--| / for k / /34 y(t) = 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - h) / / / /34 - 20 exp|-2 ln|--| (t - h)| Heaviside(t - h) / / I used maple to check my solution > laplace(50+35*exp(ln(34/35)*t)+20*Heaviside(t-2)-20*exp(ln(34/35)*(t-2))*Heaviside(t-2), t, s); print(`output redirected...`); # input placeholder / /34\ 5 |17 s - 2 (5 + 2 exp(-2 s)) ln|--|| // ------------------------------------- / /34 |-ln|--| + s| s / / Using this I verified that I did the inverse Laplace correctly. 3. ( CAS) Complete Daphne's table. In particular explain why large values of H give the game time of death > with(inttrans); print(??); # input placeholder > Y12 := 50+35*exp(2*ln(35/34)*t)+20*Heaviside(t-12)-20*exp(2*ln(35/34)*(t-12))*Heaviside(t-12); print(`output redirected...`); # input placeholder / /35 50 + 35 exp|2 ln|--| t| + 20 Heaviside(t - 12) / / / /35 - 20 exp|2 ln|--| (t - 12)| Heaviside(t - 12) / / > solve(Y12(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y11 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-11)-20*exp(-2*ln(34/35)*(t-11))*Heaviside(t-11); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 11) / / / /34 - 20 exp|-2 ln|--| (t - 11)| Heaviside(t - 11) / / > solve(Y11(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y10 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-10)-20*exp(-2*ln(34/35)*(t-10))*Heaviside(t-10); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 10) / / / /34 - 20 exp|-2 ln|--| (t - 10)| Heaviside(t - 10) / / > solve(Y10(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y9 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-9)-20*exp(-2*ln(34/35)*(t-9))*Heaviside(t-9); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 9) / / / /34 - 20 exp|-2 ln|--| (t - 9)| Heaviside(t - 9) / / > solve(Y9(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y8 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-8)-20*exp(-2*ln(34/35)*(t-8))*Heaviside(t-8); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 8) / / / /34 - 20 exp|-2 ln|--| (t - 8)| Heaviside(t - 8) / / > solve(Y8(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y7 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-7)-20*exp(-2*ln(34/35)*(t-7))*Heaviside(t-7); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 7) / / / /34 - 20 exp|-2 ln|--| (t - 7)| Heaviside(t - 7) / / > solve(Y7(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y6 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-6)-20*exp(-2*ln(34/35)*(t-6))*Heaviside(t-6); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 6) / / / /34 - 20 exp|-2 ln|--| (t - 6)| Heaviside(t - 6) / / > solve(Y6(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y5 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-5)-20*exp(-2*ln(34/35)*(t-5))*Heaviside(t-5); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 5) / / / /34 - 20 exp|-2 ln|--| (t - 5)| Heaviside(t - 5) / / > solve(Y5(t) = 98.6, t); print(`output redirected...`); # input placeholder 6.141132735 > Y4 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-4)-20*exp(-2*ln(34/35)*(t-4))*Heaviside(t-4); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 4) / / / /34 - 20 exp|-2 ln|--| (t - 4)| Heaviside(t - 4) / / > solve(Y4(t) = 98.6, t); print(`output redirected...`); # input placeholder 6.928023129 > Y3 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-3)-20*exp(-2*ln(34/35)*(t-3))*Heaviside(t-3); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 3) / / / /34 - 20 exp|-2 ln|--| (t - 3)| Heaviside(t - 3) / / > solve(Y3(t) = 98.6, t); print(`output redirected...`); # input placeholder 7.803017527 > Y2 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-2)-20*exp(-2*ln(34/35)*(t-2))*Heaviside(t-2); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 2) / / / /34 - 20 exp|-2 ln|--| (t - 2)| Heaviside(t - 2) / / > solve(Y2(t) = 98.6, t); print(`output redirected...`); # input placeholder 8.781389817 3. ( CAS) Complete Daphne's table. In particular explain why large values of H give the game time of death > with(inttrans); print(??); # input placeholder > Y12 := 50+35*exp(2*ln(35/34)*t)+20*Heaviside(t-12)-20*exp(2*ln(35/34)*(t-12))*Heaviside(t-12); print(`output redirected...`); # input placeholder / /35 50 + 35 exp|2 ln|--| t| + 20 Heaviside(t - 12) / / / /35 - 20 exp|2 ln|--| (t - 12)| Heaviside(t - 12) / / > solve(Y12(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y11 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-11)-20*exp(-2*ln(34/35)*(t-11))*Heaviside(t-11); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 11) / / / /34 - 20 exp|-2 ln|--| (t - 11)| Heaviside(t - 11) / / > solve(Y11(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y10 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-10)-20*exp(-2*ln(34/35)*(t-10))*Heaviside(t-10); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 10) / / / /34 - 20 exp|-2 ln|--| (t - 10)| Heaviside(t - 10) / / > solve(Y10(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y9 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-9)-20*exp(-2*ln(34/35)*(t-9))*Heaviside(t-9); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 9) / / / /34 - 20 exp|-2 ln|--| (t - 9)| Heaviside(t - 9) / / > solve(Y9(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y8 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-8)-20*exp(-2*ln(34/35)*(t-8))*Heaviside(t-8); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 8) / / / /34 - 20 exp|-2 ln|--| (t - 8)| Heaviside(t - 8) / / > solve(Y8(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y7 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-7)-20*exp(-2*ln(34/35)*(t-7))*Heaviside(t-7); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 7) / / / /34 - 20 exp|-2 ln|--| (t - 7)| Heaviside(t - 7) / / > solve(Y7(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y6 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-6)-20*exp(-2*ln(34/35)*(t-6))*Heaviside(t-6); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 6) / / / /34 - 20 exp|-2 ln|--| (t - 6)| Heaviside(t - 6) / / > solve(Y6(t) = 98.6, t); print(`output redirected...`); # input placeholder 5.662355357 > Y5 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-5)-20*exp(-2*ln(34/35)*(t-5))*Heaviside(t-5); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 5) / / / /34 - 20 exp|-2 ln|--| (t - 5)| Heaviside(t - 5) / / > solve(Y5(t) = 98.6, t); print(`output redirected...`); # input placeholder 6.141132735 > Y4 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-4)-20*exp(-2*ln(34/35)*(t-4))*Heaviside(t-4); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 4) / / / /34 - 20 exp|-2 ln|--| (t - 4)| Heaviside(t - 4) / / > solve(Y4(t) = 98.6, t); print(`output redirected...`); # input placeholder 6.928023129 > Y3 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-3)-20*exp(-2*ln(34/35)*(t-3))*Heaviside(t-3); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 3) / / / /34 - 20 exp|-2 ln|--| (t - 3)| Heaviside(t - 3) / / > solve(Y3(t) = 98.6, t); print(`output redirected...`); # input placeholder 7.803017527 > Y2 := 50+35*exp(-2*ln(34/35)*t)+20*Heaviside(t-2)-20*exp(-2*ln(34/35)*(t-2))*Heaviside(t-2); print(`output redirected...`); # input placeholder / /34 50 + 35 exp|-2 ln|--| t| + 20 Heaviside(t - 2) / / / /34 - 20 exp|-2 ln|--| (t - 2)| Heaviside(t - 2) / / > solve(Y2(t) = 98.6, t); print(`output redirected...`); # input placeholder 8.781389817 > h Time body moved time of death 12 "6:00 p.m." "12:20 a.m." 11 "7:00 p.m." "12:20 a.m." 10 "8:00 p.m." "12:20 a.m." 9 "9:00 p.m." "12:20 a.m." 8 "10:00 p.m." "12:20 a.m." 7 "11:00 p.m." "12:20 a.m." 6 "12:00 p.m." "12:30 a.m." 5 "1:00 a.m." "11:52 a.m." 4 "2:00 a.m." "11:04 p.m." 3 "3:00 a.m." "10:12 p.m." 2 "4:00 a.m." "9:13 p.m." The time of death is the same before 1:00 p.m because 12:20 was the time of death if it was in the fridge from the very start making it the latest possible time of death. The tempurature at a certain time is known and if the 4. Who does Daphne want to question and why? The cook because he was the only one around at both the time of death and body move time after eliminating impossible times. The victim was seen talking with the cook until his break at 10:30 so he was confirmed alive until that point and the shop closed at 2 so assuming no one came in using a spare key after the place was closed up. All times of death calculated except for two were 11 or later and those two had body move times after 2. Since the rest were 11 or later and Slim left at 11 he could have been around for the murder at 11 but would have had to be around to move the body at 2 and since that was closing time he would not have likely been able to re enter the dinner to move the body without being noticed with everything closing up. So the cook was the only one around for two of the required points of time. The wife isn't even a viable suspect at this point because no calculated time of death went that far back and if the body was moved before 12:20 then it would have been colder than the tempurature it was found at. 5. dT -- = k(T - Tm) dt T(0)=To > diff(dT/dt = k(T-Tm), T); print(`output redirected...`); # input placeholder 0 = D(k)(T - Tm) > laplace(98.6 = Tm+(To-Tm)*exp(kt), t, s); print(`output redirected...`); # input placeholder 98.60000000 Tm + (To - 1. Tm) exp(kt) ----------- = ------------------------- 1. 1. s s These were just some initial ideas to try and solve question 5 but neither were even close. The equation of the tangent line is y=mx+b where m is equal to the derivative. dT -- = k(T - Tm) dt T(0)=To so T'(0)= k(To - Tm) and since it goes through (0,To) the Y-intercept is To > T = k(To-Tm)*t+To; print(`output redirected...`); # input placeholder T = k(To - Tm) t + To "T is equal to 98.4 to subbing in give me" 98.4-To= k(To - Tm) t Dividing both sides by k(To - Tm) 98.4 - To t = ---------- k(To - Tm) This is the result shown in problem 5.Explanation / Answer
Better post screen shots...this seems impossible to follow...dont mind .. :)