PRACTICE PROBLEMS
1. A company earns revenue (income) and also makes payroll payments. Assume that
revenue is earned continuously, that payroll payments are made continuously, and that the
only factors of affecting net worth are revenue and payroll. The company’s revenue is
earned at a continuous annual rate of 5% times the net worth. At the same time, the
company’s payroll obligations are paid out at a constant rate of 200 million dollars a year.
Use this information to write a differential equation to model the net worth of the company,
𝑊, in millions of dollars, as a function of time, 𝑡, in years.
2. A patient having major surgery is given the antibiotic Vancomycin intravenously at a rate
of 85 mg per hour. The rate at which the drug is excreted from the body is proportional to
the quantity present, with proportionality constant 0.1 if time is in hours. Write a
differential equation for the quantity, 𝑄 in mg of Vancomycin in the body after 𝑡 hours.
3. Money in a bank account earns interest at a continuous annual rate of 5% times the current
balance. Write a differential equation for the balance, 𝐵, in the account as a function of
time, 𝑡, in years.
4. A bank account that initially contains $25,000 earns interest at a continuous rate of 4% per
year. Withdrawals are made out of the account at a constant rate of $2,000 per year. Write
a differential equation for the balance, 𝐵, in the account as a function of the number of
years, 𝑡.
5. Morphine is administered to a patient intravenously at a rate of 2.5 mg per hour. About
34.7% of the morphine is metabolized and leaves the body each hour. Write a differential
equation for the amount of morphine, 𝑀, in milligrams, in the body as a function of time,
𝑡, in hours.
6. A cup of coffee contains about 100 mg of caffeine. Caffeine is metabolized and leaves the
body at a continuous rate of about 17% every hour. Write a differential equation for the
amount, 𝐴, of caffeine in the body as a function of the number of hours, 𝑡, since the coffee
was consumed.
7. A person deposits money into an account at a continuous rate of $6000 a year, and the
account earns interest at a continuous rate of 7% per year. Write a differential equation for
the balance in the account, 𝐵, in dollars, as a function of years, 𝑡.
8. A pollutant spilled on the ground decays at a rate of 8% a day. In addition, cleanup crews
remove the pollutant at a rate of 30 gallons a day. Write a differential equation for the
amount of pollutant, 𝑃, in gallons, left after 𝑡 days.
9. Alcohol is metabolized and excreted from the body at a rate of about one ounce of alcohol
every hour. If some alcohol is consumed, write a differential equation for the amount of
alcohol, 𝐴 (in ounces), remaining in the body as a function of 𝑡, the number of hours since
the alcohol was consumed.
10. Toxins in pesticides can get into the food chain and accumulate in the body. A person
consumes 10 micrograms a day of a toxin, ingested throughout the day. The toxin leaves
the body at a continuous rate of 3% every day. Write a differential equation for the amount
of toxin, 𝐴, in micrograms, in the person’s body as a function of the number of days, 𝑡.
11. The rate at which a drug leaves the bloodstream and passes into the urine is proportional
to the quantity of the drug in the blood at that time. If an initial dose of 𝑄0 is injected
directly into the blood, 20% is left in the blood after 3 hours. Write and solve a differential
equation for the quantity, 𝑄, of the drug in the blood after 𝑡 hours.
12. Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed,
the quantity of drug in the body decreases at a rate proportional to the amount left in the
body. The half-life of hydrocodone bitartrate in the body is 3.8 hours and the dose is 10
mg. Write a differential equation for the quantity, 𝑄, of hydrocodone bitartrate in the body
at time t, in hours since the drug was fully absorbed.
13. A patient is given the drug warfarin, an anticoagulant, intravenously at the rate of 0.5
mg/hour. Warfarin is metabolized and leaves the body at the rate of about 2% per hour.
Write a differential equation for the quantity, 𝑄 (in mg), of warfarin in the body after 𝑡
hours.
14. A bank account that earns 10% interest compounded continuously has an initial balance
of zero. Money is deposited into the account at a continuous rate of $1000 per year. Write
a differential equation that describes the rate of change of the balance 𝐵 = 𝑓(𝑡).
15. The amount of ozone, 𝑄 , in the atmosphere is decreasing at a rate proportional to the
amount of ozone present. If time 𝑡 is measured in years, the constant of proportionality is
0.0025. Write a differential equation for 𝑄 as a function of 𝑡.