Mathematis II Tutorial 5 Basi mathematial modelling Groups: B03 & B08 February 29, 2012 Mathematis II Ngo Quo Anh Ngo Quo Anh Department of Mathematis National University of Singapore 1/13
: The ost of making ans We setup notations first: J is the ost of aluminium per square m, K is the ost per m of welding. Regarding to J, we need to find total area of the an; and for K we need to find the length of welding. From the question, only the top of the an is welded on. Thus, the length of the welding is nothing but the perimeter of the top, whih equals 2πr. For the area of the an, it is simply equal to twie of the area of the top plus the area of the side of the an. Mathematially, we find that 2r = 5 m h = 12.5 m Mathematis II Ngo Quo Anh Total area of the an = 2πr 2 + 2V r. In onlusion, we find the ost as The ost Cr) = J 2πr 2 + 2V r ) + K2πr). 2/13
: The ost of making ans In order to minimize Cr), we have to find its ritial points. By differentiating, we first have C r) = J 4πr 2V ) r 2 + 2πK ) = J 4πr 2πr2 h r 2 + 2πK = J 4πr 2πh) + 2πK. By studying C r) = 0, we get that Jh 2r) = K. Equivalently, we have Mathematis II Ngo Quo Anh h r = 2 + K/J. r This tells us that if r is fixed, both h and K J have the same monotoniity. By using the following fats h = 12.5, r = 2.5, we get that K J = 7.5. 3/13
: The Malthusian growth model Suppose P is the population, B is the birth-rate, and D is the death-rate. The Malthusian growth model The model an be stated as a simple ODE Equivalently, we have d P t) = B D)P t). dt P t) = P 0 e B D)t, Mathematis II Ngo Quo Anh where P 0) = P 0 is the initial population. Using this model with P 0) = 10, 000 and the fat that P 1 + 1 2 ) = 11, 000 we find that 11, 000 = 10, 000e 3 2 B D). 4/13
: The Malthusian growth model Solving the equation above, we get that B D = 2 3 ln 11 10. To measure the number of bateria after 10 hours, we simply alulate P 10) whih is P 10) = 10, 000e 2 3 ln 11 10 )10 14, 600. To find the duration so that the number of bateria reahes 20, 000, we simply solve for t from the following Mathematis II Ngo Quo Anh 20, 000 = 10, 000e 2 3 ln 11 10 )t. Obviously, t 18.18 hours). NB: The Malthusian growth model is the diret anestor of the logisti funtion. 5/13
: The logisti growth model Sine a fixed number of olonists are sent out eah year, the rate of emigration is nothing but onstant. For that reason, let us assume that is the number of emigrants per year. Sine the Malthusian model would hold if there were no emigration, we have the following ODE B D + d P t) = B D)P t). dt By solving, using P t) = P h t) + A, we get that P t) = P 0) B D ) e B D)t, t 0, where P 0) denotes the initial population. Depending on the sign of P 0) B D, we basially have three ases If P 0) < B D, then P t) 0 exponentially. If P 0) = B D, then P t) = B D all the time. If P 0) >, then P t) + exponentially. B D Mathematis II Ngo Quo Anh 6/13
: The logisti growth model If the rate of emigration is proportional to time, say t. The orresponding ODE is now d P t) = B D)P t) t. dt Using the method of undetermined oeffiients, the general solution P t) = P h t) + At + B) is ) P t) = B D t+ B D) 2 + P 0) B D) 2 e B D)t, where t 0 and P 0) is the initial population. Again, we basially have three ases If P 0) <, then P t) and then P t) 0 B D) 2 sine the exponential funtions grow faster than linear funtions. If P 0) =, then P t) = B D) 2 B D t + linearly. If P 0) > B D) 2 + B D) 2, then P t) + exponentially. Mathematis II Ngo Quo Anh 7/13
: The logisti growth model Sine the birth-rate B and death-rate D are onstants, we an adopt the Malthusian growth model to the problem, that is, we have d P t) = B D)P t). dt The general solution is Sine P 20) P 0) P t) = P 0)e B D)t, t 0. = 2, we get that 2 = e B D)20 whih implies that B D = ln 2 20. After the departure of the women, say at t = t, learly B is zero. Therefore, Using the fat that P t) = P t)e Dt t), t t. P t + 10) = 1 2 P t), Mathematis II Ngo Quo Anh 8/13
: The logisti growth model we find that e 10D = 1 ln 2 2. Hene, D = 10. Having D, we an solve for B from the equation B D = ln 2 20. A simple alulation shows that B = 0.10397, that is, 10.397% per year. From the solution above, one an find that the following assumptions have been made: Men and old women have same death rate as young women, whih is not true in reality beause men usually smoke, get into fights, et while on the other hand old women are indestrutible. The death rate of the remaining population is not hanged by the departure of the women. That is D before t and D after t are the same. This helps us to go bak to solve for B one we have D. This is a very questionable assumption sine morale will be affeted, et. Question: Study the presentation of P t) using t = 0, t. Mathematis II Ngo Quo Anh 9/13
: The logisti growth model Let N be the number of neutrons. It is known that D is a positive onstant and B = sn for some onstant s > 0. The model an be stated via the following ODE To solve this, we first write dt = d Nt) = sn D)Nt). dt dn sn D)N = s D By integrating, we get that 1 sn D 1 sn ) dn. t + C = 1 D ln sn D 1 D ln N = 1 D ln sn D N In other words, sn D N = edt+c), or Nt) = D s e Dt+C).. Mathematis II Ngo Quo Anh 10/13
: The logisti growth model Apparently, we have to determine the orret sign for Nt). This proess depends on the sign of sn0) D. Suppose sn0) D > 0. It follows from N0) = D s e DC > D s that is the orret sign. Thus showing that Nt) = D s e Dt+C) Mathematis II Ngo Quo Anh Suppose sn0) D = 0. By the no-rossing priniple, we find that N = D s is a solution. Suppose sn0) D < 0. We then find that the solution is Nt) = D s + e Dt+C). 11/13
: The logisti growth model First, remember the phase portrait of the logisti equation dp dt = BP sp 2, i.e., the lassifiation of solutions. Initially, you have 200 bugs in a bottle but 2 days later, you have 360 bugs. As suh, the solution for the logisti growth model was inreasing. Consequently, the form of suh a solution is B P t) = ), t 0. s + B P 0) s e Bt From the question, there hold B = 3 2, P 0) = 200. We still need to find s. Thanks to P 2) = 360, we find that Therefore, 3/2 360 = ). s + 3/2 200 s e 3/2)2 s = 1 9e 3 5 1200 e 3 1 0.3992 Mathematis II Ngo Quo Anh 12/13
: The logisti growth model Thus, after 3 days, the number of bugs is equal to P 3) whih is P 3) = 3/2 1 9e 3 5 1200 e 3 1 + 3/2 200 1 1200 ) 372. 9e 3 5 e e 3 1 3/2)3 Conerning the number of bugs that eventually have, say P, we atually study the following limit lim P t) = t + lim t + B ) s + B P 0) s e Bt Mathematis II Ngo Quo Anh whih is nothing but B s. Therefore, P = B s = 376. In our ase, the birth rate B = 3 2 seems to be high. 13/13