1. A certain small country has $56 billion in paper currency in circulation, and each day...

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1. A certain small country has $56 billion in paper currency in circulation, and each day $152 million comes into the country's banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let x = : x(t) denote the amount of new currency (in billions of dollars) in circulation at time t (in days), with x(0) = 0. (i) Formulate a mathematical model in the form of an initial value problem that represents the "flow" of the new currency into circulation. Hint: Think about this problem as a mixing problem. For the new currency, what is the "rate out", and what is the flow and concentration for "rate in? (ii) Solve the initial-value problem found in part (a). (iii) Using your solution, show that it will take about 3.54 years to replace 97% of the old currency. 2. There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor (1 m/P). The modified logistic model is given by the differential equation dP dt = kP (1-1) (1-7) P M m P (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P 1. A certain small country has $56 billion in paper currency in circulation, and each day $152 million comes into the country's banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let x = : x(t) denote the amount of new currency (in billions of dollars) in circulation at time t (in days), with x(0) = 0. (i) Formulate a mathematical model in the form of an initial value problem that represents the "flow" of the new currency into circulation. Hint: Think about this problem as a mixing problem. For the new currency, what is the "rate out", and what is the flow and concentration for "rate in? (ii) Solve the initial-value problem found in part (a). (iii) Using your solution, show that it will take about 3.54 years to replace 97% of the old currency. 2. There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor (1 m/P). The modified logistic model is given by the differential equation dP dt = kP (1-1) (1-7) P M m P (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P