1. [-I2 Points] DETAILS ZILLDIFFEQMODAP11 1.3.001. MY NOTES ASK YOUR TEACHER Assume that in the absence of immigration and emigration, the growth of a country's population P(t) satisfies dP/dt = kP for some constant k > 0 (see Equation (1) of Section 1.3). Determine a differential equation governing the growing population P(t) of the country when individuals are allowed to immigrate into the country at a constant rate r > 0. (Use P for P(t).) 019 = dt What is the differential equation for the population P(t) of the country when individuals are allowed to emigrate at a constant rate r > 0? 0'19 =' ' dt Need Help? 2. [I1 Points] DETAILS ZILLDIFFEQMODAP11 1.3.002. MY NOTES ASK YOUR TEACHER The population model given in (1) in Section 1.3 do a P or g = kP (1) dt or fails to take death into consideration; the growth rate equals the birth rate. In another model ofa changing population of a community it is assumed that the rate at which the population changes is a net ratethat is, the difference between the rate of births and the rate of deaths in the community. Determine a model for the population P(t) if both the birth rate and the death rate are proportional to the population present at time t > 0. (Assume the constants of proportionality for the birth and death rates are k1 and k2 respectively. Use Pfor P(t).) dP _' ' E _ Need Help? 3. [-I3 Points] DETAILS ZILLDIFFEQMODAP11 1.3.005. MY NOTES ASK YOUR TEACHER A cup of coffee cools according to Newton's law of cooling (3) in Section 1.3. dT dT .. T T or = k T T 3 dt m d: ( m) ( ) Use data from the graph of the temperature T{t) in the figure to estimate the constants Tm, To, and k in a model of the form of a first-order initial-value problem: dT/dt = k(T Tm), T(0) = To- T 200 150 100 50 minutes 4. [-11 Points] DETAILS Zl LLDIFFEQMODAP11 1.3.007. MY NOTES ASK YOURTEACHER PRACTICE ANOTHER Suppose a student carrying a u virus returns to an isolated college campus of 7000 students. Determine a differential equation governing the number of students x(t) who have contracted the u if the rate at which the disease spreads is proportional to the number of interactions between students with the u and students who have not yet contracted it. (Use k > 0 for the constant of proportionality and x for x(t).) dt Need Help? 5. [-12 Points] DETAILS ZILLDIFFEQMODAPH 1.3.008. MY NOTES ASK YOUR TEACHER At a time denoted as t = 0 a technological innovation is introduced into a community that has a xed population of 11 people. Determine a differential equation for the number of people x(t) who have adopted the innovation at time t if it is assumed that the rate at which the innovations spread through the community is jointly proportional to the number of people who have adopted it and the number of people who have not adopted it. (Use k > 0 for the constant of proportionality and x for x(t). Assume that initially one person adopts the innovation.) dx E x(0) = ' ' Need Help? 6. [-12 Points] DETAILS ZILLDIFFEQMODAP11 1.3.009. MY NOTES ASK YOURTEACHER PRACTICE ANOTHER Suppose that a large mixing tank initially holds 500 gallons of water in which 30 pounds of salt have been dissolved. Pure water is pumped into the tank at a rate of 5 gal/minr and when the solution is well stirred, it is then pumped out at the same rate. Determine a differential equation for the amount of salt A(t) in the tank at time t > 0. What is Am)? (Use A for A(t).) 2-: dr Need Help? 7. [-I1 Points] DETAILS Zl LLDIFFEQMODAP11 1.3.010.MI. MY NOTES ASK YOURTEACHER PRACTICE ANOTHER Suppose that a large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min. 1f the concentration of the solution entering is 2 Ib/galr determine a differential equation (in Ib/min) for the amount of salt A(t) (in lb) in the tank at time t > 0. (Use A for A(t).) = lb/min cit Need Help