Fall 2005
Practice Exam 3
Kingston 11/15/2005
1. (a) Find the general solution to the differential equation
y ' = ( x2 +1) y
(b) Solve the initial value problem
y ' = ( x2 +1) y, y(0) = 2.
2.
(a) Find the general solution to the differential equation
y ' = ( 2 x +1) (y + 2).
(b) Solve the initial value problem
y ' = ( 2 x +1) (y + 2), y(0) = 1.
3. A population of insects at time t, Q(t) increases at a rate proportional to the size of the population, with the coefficient of proportionality 0.024. The time t is measured in hours.
(a) Write a differential equation for the function Q(t).
(b) Find the general solution to this equation.
(c) If the initial population is 700 , how many insects will be present after 14 hours ?
4. A yam is put in a 220 o C oven and heats up according to the Newton's Law of Heating.
(a) If the yam is at 22 o C when it is put in oven, write a differential equation satisfied by the temperature, H(t).
(b) Solve the differential equation in (a) using the fact after 20 minutes the temperature of the yam is 100 o C.
(c) Calculate the temperature of the yam after 35 minutes
5. The amount of a drug in a patient's bloodstream, Q(t), in mg decreases at a rate proportional at each instant to the amount of the drug present with the coefficient of proportionality -0.14. The time t, is measured in hours.
(a) Write a differential equation for the function Q(t).
(b) Find the general solution to the equation.
(c) If the initial amount Q(0)=500, how much drug is left after 20 hours ?
6. The following system of differential equations represents the interaction between two populations, x and y.
dx/dt = x + 0.5 x y
dx/dt = -y + 0.4 x y
(a) Describe how the species interact. Are they helpful or harmful to each other ?
(b) Write a differential equation for dy/dx .
7. A radioactive isotope is released into the air as an
industrial by-product. This isotope is not very stable due to
radioactive decay. Twenty percent of the original radioactive material
loses its radioactivity after each month. If 70 grams
of this isotope are released into the atmosphere at the end of the
first and every subsequent month, then
(a) how much radioactive material is in the atmosphere at the end of the tenth month?
(b) In the long run, i.e., if the situation goes on forever, what will be the amount of this radioactive isotope in the atmosphere at the end of each month ?
8. Each day at noon a hospital patient receives 20 mg of a drug. This drug is gradually eliminated from the system in such a way that over each 24 hour period the amount of the drug from the original dose is reduced by 20 %. On the other hand, the patient is receiving a new 20 mg dose each day.
(a) How much of the drug is present in the patient's system immediately after noon on the eighth day of treatment ?
(b) If the regime is continued indefinitely, at what level will the amount of the drug in the patient's system stabilize, measured immediately after noon every day ?
9. Use geometric series to find simple fractions that equal the following repeating decimals:
(a) A = 0.27777...
(b) B = 0.27272727...
10. At the end of year 2000
world oil reserves were about 1000 billion barrels. During
2000,
about 27 billion barrels of oil were consumed. Over the past decade, oil
consumption has been increasing at about 1% per
year *.
(a) Assuming oil consumption increases at this rate in the future, how long will the reserves last ?
(b) Assuming oil consumption increases at the rate of 0.8% per year in the future, how long will the reserves last ?
*. According to www.bp.com/energy/world_stat_rev/oil/reserves.asp