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7.5. MODELING WITH DIFFERENTIAL EQUATIONS
Activity 7.11.
Suppose you have a bank account that grows by 5% every year. Let A(t) be the amount
of money in the account in year t.
(a) What is the rate of change of A with respect to t?
(b) Suppose that you are also withdrawing $10,000 per year. Write a differential
equation that expresses the total rate of change of A.
(c) Sketch a slope field for this differential equation, find any equilibrium solutions,
and identify them as either stable or unstable. Write a sentence or two that
describes the significance of the stability of the equilibrium solution.
(d) Suppose that you initially deposit $100,000 into the account. How long does it
take for you to deplete the account?
(e) What is the smallest amount of money you would need to have in the account
to guarantee that you never deplete the money in the account?
(f) If your initial deposit is $300,000, how much could you withdraw every year
without depleting the account?
⊳
Activity 7.12.
A dose of morphine is absorbed from the bloodstream of a patient at a rate proportional
to the amount in the bloodstream.
(a) Write a differential equation for M(t), the amount of morphine in the patient’s
bloodstream, using k as the constant proportionality.
(b) Assuming that the initial dose of morphine is M 0 , solve the initial value problem
to find M(t). Use the fact that the half-life for the absorption of morphine is
two hours to find the constant k.
(c) Suppose that a patient is given morphine intraveneously at the rate of 3 milligrams per hour. Write a differential equation that combines the intraveneous
administration of morphine with the body’s natural absorption.
(d) Find any equilibrium solutions and determine their stability.
(e) Assuming that there is initially no morphine in the patient’s bloodstream, solve
the initial value problem to determine M(t). What happens to M(t) after a very
long time?
(f) To what rate should a doctor reduce the intraveneous rate so that there is
eventually 7 milligrams of morphine in the patient’s bloodstream?
⊳
7.5. MODELING WITH DIFFERENTIAL EQUATIONS
Activity 7.11.
Suppose you have a bank account that grows by 5% every year. Let A(t) be the amount
of money in the account in year t.
(a) What is the rate of change of A with respect to t?
(b) Suppose that you are also withdrawing $10,000 per year. Write a differential
equation that expresses the total rate of change of A.
(c) Sketch a slope field for this differential equation, find any equilibrium solutions,
and identify them as either stable or unstable. Write a sentence or two that
describes the significance of the stability of the equilibrium solution.
(d) Suppose that you initially deposit $100,000 into the account. How long does it
take for you to deplete the account?
(e) What is the smallest amount of money you would need to have in the account
to guarantee that you never deplete the money in the account?
(f) If your initial deposit is $300,000, how much could you withdraw every year
without depleting the account?
⊳
Activity 7.12.
A dose of morphine is absorbed from the bloodstream of a patient at a rate proportional
to the amount in the bloodstream.
(a) Write a differential equation for M(t), the amount of morphine in the patient’s
bloodstream, using k as the constant proportionality.
(b) Assuming that the initial dose of morphine is M 0 , solve the initial value problem
to find M(t). Use the fact that the half-life for the absorption of morphine is
two hours to find the constant k.
(c) Suppose that a patient is given morphine intraveneously at the rate of 3 milligrams per hour. Write a differential equation that combines the intraveneous
administration of morphine with the body’s natural absorption.
(d) Find any equilibrium solutions and determine their stability.
(e) Assuming that there is initially no morphine in the patient’s bloodstream, solve
the initial value problem to determine M(t). What happens to M(t) after a very
long time?
(f) To what rate should a doctor reduce the intraveneous rate so that there is
eventually 7 milligrams of morphine in the patient’s bloodstream?
⊳