1IntrotomathmodelingThis exercise is adapted from E. A.Bender, An introduction to mathematical modeling,Dover, New York, 2000.We want to device a strategy to minimize the cost of forest fires. Let B(t) be the area offorest burnt by time t, where time is measured from the time of detection, t = 0. We assumethat the fire has stopped when B'(t) = 0. Let T, be the time the fire is first attacked and T, thetime it is brought under control, i.e T, is the least t > 0 such that B'(t) = 0. Let x be the sizeof the fire fighting force (assumed constant from Ta to Te). The cost for fighting a particularfire are:Cb; the cost per acre of fire (burnt acreage plus cleanup expanses)Cx; the cost in support and salary per fire fighter per unit time.Cs; one-shot costs per fire fighter (such as transportation to and from the sites)Ch; costs per unit time, while the fire is burning, for maintaining the organization on anemergency basis, redirecting traffic, and so on.Thus, thetotal cost isC = C,B(T) +(xCx +C)(T。- Ta) + xCs:To minimize C as a function of x, we must determine B(t). We assume that each fire fighterreduces the burning rate of the fire at a constant rate E, that is, decreases B"(t) by E. Thus[ B'(t) = b(t)fort His required if the fire is ever going to be stopped. What does this mean?2 Solve (1) to obtain B(tb), and show that this leads to the following expression for C:C = Co + C,z + [(HC /E) + C + (Cib/2)]baEzwhere Co is a constant, ba = b(Ta) = G + HTa, and z = x - H/E
Intro to math modeling 1 This exercise is adapted from E. A. Bender, An introduction to mathematical modeling, Dover, New York, 2000. We want to device a strategy to minimize the cost of forest fires. Let B(t) be the area of forest burnt by time t, where time is measured from the time of detection, t = 0. We assume that the fire has stopped when B 0 (t) = 0. Let Ta be the time the fire is first attacked and Tc the time it is brought under control, i.e Tc is the least t > 0 such that B 0 (t) = 0. Let x be the size of the fire fighting force (assumed constant from Ta to Tc). The cost for fighting a particular fire are: Cb ; the cost per acre of fire (burnt acreage plus cleanup expanses). Cx; the cost in support and salary per fire fighter per unit time. Cs ; one-shot costs per fire fighter (such as transportation to and from the sites). Ct ; costs per unit time, while the fire is burning, for maintaining the organization on an emergency basis, redirecting traffic, and so on. Thus, the total cost is C = CbB(Tc) + (xCx + Ct)(Tc − Ta) + xCs . To minimize C as a function of x, we must determine B(t). We assume that each fire fighter reduces the burning rate of the fire at a constant rate E, that is, decreases B 00(t) by E. Thus ( B 0 (t) = b(t) for t H is required if the fire is ever going to be stopped. What does this mean? 2 Solve (1) to obtain B(tb ), and show that this leads to the following expression for C: C = C0 + Csz + [(HCx/E) + Ct + (Cbba/2)]ba Ez where C0 is a constant, ba = b(Ta) = G + HTa, and z = x − H/E
2Midterm3UsetheexpressionforCtoderivetheoptimalvalueforx:Ca+2Ct/ba+2HCx/EbaHX+=ba1E2C,E4 Discuss inwordstheimplicationsofthemodel, criticizethe conclusionsthat seemsunrealisticand theassumptions thatled totheseconclusions,etc.5 The model does not take into account human cost related to injuries or deaths of thefirefighters.How would yu include sucha cost in themodel?
2 Midterm 3 Use the expression for C to derive the optimal value for x: x? = ba s Ca + 2Ct/ba + 2HCx/Eba 2CsE + H E 4 Discuss in words the implications of the model, criticize the conclusions that seems unrealistic and the assumptions that led to these conclusions, etc. 5 The model does not take into account human cost related to injuries or deaths of the firefighters. How would yu include such a cost in the model?