Honors 2201-4
Homework 10 Extra Credit: due Wednesday October 29
A correct solution will add 5 points to your homework score.
Fall 2003
Lead-210 (210 P b) is a radioactive substance with a half-life of 22 years. Let
y(t) be the amount of 210 P b in a sample at time t, with y(0) = y0 .
(a) What is the initial value problem which models the rate of decay of 210 P b?
(b) Solve the initial value problem of part (a) to obtain a formula for y(t).
In particular, determine the value of the decay constant k > 0 which appears
in the initial value problem. (Hint: Use the half-life of 210 P b.)
210
P b occurs primarily in nature as a byproduct of the breakdown of the
radioactive substance radium-226 (226 Ra). Although 226 Ra decays to 210 P b,
this is a slow process since the half life of 226 Ra is 1600 years. Hence, over
smaller time spans, we may assume that the amount of 226 Ra present, and
hence its rate of decay r, remain constant. Since the disintegration of a 226 Ra
atom will eventually yield a 210 P b atom, the initial value problem modeling
the decay of 210 P b in a sample containing 226 Ra must be changed to:
the rate of change of
present.
210
P b is equal to r plus (−k) times the amount of
210
Pb
(c) Formulate and solve this new initial value problem for y = y(t), the
amount of 210 P b present in a sample at time t. Continue to assume that
y(0) = y0 .
Both 210 P b and 226 Ra are found in all oil paintings since they are contained
in “white lead,” a pigment used by artists for over 2000 years. However, in
the process that is used to produce the “white lead” from lead ore, 90% to
95% of the radium is removed while all of the lead remains. This sets up an
imbalance which is not corrected until a corresponding 90 − 95% of the 210 P b
disintegrates. This process can be used to give information on the age of a
painting. In particular, it can be used to detect art forgeries.
What follows is a true story. The “Disciples at Emmaus” was believed to be
an authentic Vermeer, approximately 300 years old. In fact, the Rembrandt
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Society paid $170,000 for the painting. However, the authenticity of the
painting and a number of other purported Vermeers such as the “Lace maker”
were called into question. The major tool used to test the paintings was a
radon/lead analysis as now follows:
(d) Rewrite your solution for y(t) in part (c) in the form ky0 = ....
The quantity ky0 is the rate of decay of 210 P b at time t = 0. We will take
y = y(t) to be the amount of 210 P b in an oil painting t years after it is
completed; hence ky0 is the rate of decay of 210 P b when the painting is brand
new. It is known that ky0 cannot be more than 20, 000dpm (disintegrations
per minute) per gram of white lead; in general it should be considerably less
than 20, 000dpm per gram.
(e) Measurements on the “Disciples of Emmaus” gave r = 0.8dpm and
ky = 8.5dpm per gram of white lead. Can this be an original 300 year
old Vermeer painting? Did the Rembrandt Society make a wise investment?
(Hint: Compute ky0 from your formula in part (d).) Repeat your analysis
on the “Lace Maker,” where the measurements were r = 1.4 and ky = 1.5.
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