Question

*20 On pp. 188-189 of ODELA we learn about a 1st order linear DE system model for quantities of lead stored in 1) the blood, 2) body tissues, and 3) bones. The model is [x] [x] [Iz(t)] [-13/360 272/21875 7/200000 * = A x2 + 0 , where A = 1/90 -1/35 0 [23] [X3] [0] 7/18000 -7/200000 (a) Approximate eigenpairs for the matrix A are provided at the top of p. 190. Use them to write the homogeneous solution—i.e., the solution in the case the influx into the bloodstream of lead from the environment (t) = 0. (b) If I (t) = 0 for a person previously poisoned with lead, what aspect of the model or solution indicates that the lead will be flushed out over time? MATH 231 Hand-Checked Assignment #3 (c) Continue assuming that I (t) = 0, but suppose we have initial conditions x:(0) = 50, X2(0) = 0 and X30) = 0; that is, we start with 50 units of lead in the blood and none in tissue nor bone. Solve the (homogeneous) IVP, and use it to write a formula for the amount x;(t) of lead in the bones. Find the approximate time + (in days) at which the level of lead in the bones is at its peak value. [Give your answer accurate to the tenths place.] Also, find the approximate time, following that peak, when the lead level in the bones has receded to no more than 0.5 units.

Answer 1

Page- Que:) buven that, 2.72 21875 200000 & -Ax t o where A = 1 1800 200000 Sod) Let us consides, (a, &b) Here, 2, = -4 47x102; V, - / -0.69 | -0.09 22 = -2x102 von 2 - XTO V . -0.19 23 = -3.06410S, V3 = 11 10:39 ST 892.56 Hene, 2= cest vit Gehste Vet G & ?3? Vz X = 6,8 - 4:47X102t (1) 1-0.69 -0.09 13 (-0.12 the-3-06x10-5ti 0.39 (892.56) c.) het 2, (0) 250 ; &2(0)=0, & (0) = 0 From Initial conditione (sol / C / . = -0.696, +/ 1.36 70.39€ 1-0.096) (-0.1962) 1892.56 (3) -DC2 Scanned with CamScanner

Put; G = 32.66 , (q 217.33, C3 20.00698 Page - 2 Here, x, = 32.66 e 0.844764 17:33 e-0.02 +0.006986 -0.00003 22=-22:53 € -0600472 +22.53 e 0°02t 0.00276 0.0000 3t &z=-2.94 e 0.64476 +(-3.24) e-tro2t +6.236 -0.000003t C Scanned with CamScanner