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7.5. MODELING WITH DIFFERENTIAL EQUATIONS
(e) How long does it take to reach 50% of the terminal velocity?
3. During the first few years of life, the rate at which a baby gains weight is proportional
to the reciprocal of its weight.
(a) Express this fact as a differential equation.
(b) Suppose that a baby weighs 8 pounds at birth and 9 pounds one month later.
How much will he weigh at one year?
(c) Do you think this is a realistic model for a long time?
4. Suppose that you have a water tank that holds 100 gallons of water. A briny solution,
which contains 20 grams of salt per gallon, enters the tank at the rate of 3 gallons per
minute.
At the same time, the solution is well mixed, and water is pumped out of the tank at
the rate of 3 gallons per minute.
(a) Since 3 gallons enters the tank every minute and 3 gallons leaves every minute,
what can you conclude about the volume of water in the tank.
(b) How many grams of salt enters the tank every minute?
(c) Suppose that S(t) denotes the number of grams of salt in the tank in minute t.
How many grams are there in each gallon in minute t?
(d) Since water leaves the tank at 3 gallons per minute, how many grams of salt
leave the tank each minute?
(e) Write a differential equation that expresses the total rate of change of S.
(f) Identify any equilibrium solutions and determine whether they are stable or
unstable.
(g) Suppose that there is initially no salt in the tank. Find the amount of salt S(t)
in minute t.
(h) What happens to S(t) after a very long time? Explain how you could have
predicted this only knowing how much salt there is in each gallon of the briny
solution that enters the tank.
7.5. MODELING WITH DIFFERENTIAL EQUATIONS
(e) How long does it take to reach 50% of the terminal velocity?
3. During the first few years of life, the rate at which a baby gains weight is proportional
to the reciprocal of its weight.
(a) Express this fact as a differential equation.
(b) Suppose that a baby weighs 8 pounds at birth and 9 pounds one month later.
How much will he weigh at one year?
(c) Do you think this is a realistic model for a long time?
4. Suppose that you have a water tank that holds 100 gallons of water. A briny solution,
which contains 20 grams of salt per gallon, enters the tank at the rate of 3 gallons per
minute.
At the same time, the solution is well mixed, and water is pumped out of the tank at
the rate of 3 gallons per minute.
(a) Since 3 gallons enters the tank every minute and 3 gallons leaves every minute,
what can you conclude about the volume of water in the tank.
(b) How many grams of salt enters the tank every minute?
(c) Suppose that S(t) denotes the number of grams of salt in the tank in minute t.
How many grams are there in each gallon in minute t?
(d) Since water leaves the tank at 3 gallons per minute, how many grams of salt
leave the tank each minute?
(e) Write a differential equation that expresses the total rate of change of S.
(f) Identify any equilibrium solutions and determine whether they are stable or
unstable.
(g) Suppose that there is initially no salt in the tank. Find the amount of salt S(t)
in minute t.
(h) What happens to S(t) after a very long time? Explain how you could have
predicted this only knowing how much salt there is in each gallon of the briny
solution that enters the tank.