7.4. SEPARABLE DIFFERENTIAL EQUATIONS
425
Exercises
1. The mass of a radioactive sample decays at a rate that is proportional to its mass.
(a) Express this fact as a differential equation for the mass M(t) using k for the
constant of proportionality.
(b) If the initial mass is M 0 , find an expression for the mass M(t).
(c) The half-life of the sample is the amount of time required for half of the mass
to decay. Knowing that the half-life of Carbon-14 is 5730 years, find the value
of k for a sample of Carbon-14.
(d) How long does it take for a sample of Carbon-14 to be reduced to one-quarter
its original mass?
(e) Carbon-14 naturally occurs in our environment; any living organism takes in
Carbon-14 when it eats and breathes. Upon dying, however, the organism no
longer takes in Carbon-14.
Suppose that you find remnants of a pre-historic firepit. By analyzing the
charred wood in the pit, you determine that the amount of Carbon-14 is only
30% of the amount in living trees. Estimate the age of the firepit. 5
2. Consider the initial value problem
dy
dt
= −
t
y
, y(0) = 8
(a) Find the solution of the initial value problem and sketch its graph.
(b) For what values of t is the solution defined?
(c) What is the value of y at the last time that the solution is defined?
(d) By looking at the differential equation, explain why we should not expect to
find solutions with the value of y you noted in (c).
3. Suppose that a cylindrical water tank with a hole in the bottom is filled with water.
The water, of course, will leak out and the height of the water will decrease. Let h(t)
denote the height of the water. A physical principle called Torricelli’s Law implies that
the height decreases at a rate proportional to the square root of the height.
(a) Express this fact using k as the constant of proportionality.
(b) Suppose you have two tanks, one with k = −1 and another with k = −10. What
physical differences would you expect to find?
5 This approach is the basic idea behind radiocarbon dating.
425
Exercises
1. The mass of a radioactive sample decays at a rate that is proportional to its mass.
(a) Express this fact as a differential equation for the mass M(t) using k for the
constant of proportionality.
(b) If the initial mass is M 0 , find an expression for the mass M(t).
(c) The half-life of the sample is the amount of time required for half of the mass
to decay. Knowing that the half-life of Carbon-14 is 5730 years, find the value
of k for a sample of Carbon-14.
(d) How long does it take for a sample of Carbon-14 to be reduced to one-quarter
its original mass?
(e) Carbon-14 naturally occurs in our environment; any living organism takes in
Carbon-14 when it eats and breathes. Upon dying, however, the organism no
longer takes in Carbon-14.
Suppose that you find remnants of a pre-historic firepit. By analyzing the
charred wood in the pit, you determine that the amount of Carbon-14 is only
30% of the amount in living trees. Estimate the age of the firepit. 5
2. Consider the initial value problem
dy
dt
= −
t
y
, y(0) = 8
(a) Find the solution of the initial value problem and sketch its graph.
(b) For what values of t is the solution defined?
(c) What is the value of y at the last time that the solution is defined?
(d) By looking at the differential equation, explain why we should not expect to
find solutions with the value of y you noted in (c).
3. Suppose that a cylindrical water tank with a hole in the bottom is filled with water.
The water, of course, will leak out and the height of the water will decrease. Let h(t)
denote the height of the water. A physical principle called Torricelli’s Law implies that
the height decreases at a rate proportional to the square root of the height.
(a) Express this fact using k as the constant of proportionality.
(b) Suppose you have two tanks, one with k = −1 and another with k = −10. What
physical differences would you expect to find?
5 This approach is the basic idea behind radiocarbon dating.