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\centerline{\bf Honors Calculus  166}
\centerline{\bf Spring 1992}
\centerline{\bf Homework 2}
\centerline{\bf Due January 31}
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{\bf A.} From the book:
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{\bf Page 325:} $1_{i,iv,vi,viii,x},2_{i,iv}, 3, 4_{a,c,d,e}, 5_{i,iii,vi}
 7, 8, 9, 12, 16_{a,b,c}, 19, 23, 24, 29 $.
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{\bf B.} Find the general solution to the following first order linear
differential equations:
$$1.)\, y'-2xy=x \qquad 2.)\,  xy'-y=x,\ x>0 \qquad 3.)\, y'+\cos x y'=\cos x.$$ 

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{\bf C.} Assuming that the population of the Earth now is 5 billion end that 
it grows at rate proportional to the current value with constant of
proportionality equal to 0.02, find the population of the Earth at the year
2017. Take $e^{0.5} \approxeq 1.6487$.
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{\bf D.}  A thirty years old person opens an Individual Retirement Account (IRA)
with  an initial investment of $\$ 3,000 $ and then makes anual investments of $\$
4,000$ thereafter continuously. Assuming an interest rate of $ 8\% $ find the
balance in the IRA at any time $t$. Also find the balance in the IRA when the
person will be 60 years old. It is given that $e^{2.4} \approxeq 11.023 $.
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{\bf D.} (From Boyce and Diprima Differential Equations book)
{\bf 1.}  Radium-226 has a half-life of 1620 years.  Find the time period during
which a body of this material is reduced to three-fourths of its original size.
{\bf 2.}  {\bf Radiocarbon Dating.}  An important tool in archeological research
is radiocarbon dating.  This is a means of determining the age of certain
wood and plant remains, hence of animal or human bones or artifacts found buried at
the same levels.  The procedure was developed by the American chemist Willard Libby
(1908-1980) in the early 1950's and resulted in his winning the Nobel prize for
chemistry in 1960.  Radiocarbon dating is based on the fact that some wood or
plant remains contain residual amounts of carbon-14, a radioactive isotope of
carbon.  This isotope is accumulated during the lifetime of the plant and begins
to decay at its death.  Since the half-life of carbon-14 is long (approximately
5568 years$^{5}$), measurable amounts of carbon-14 remain after many thousands of
years.  Libby showed that if even a tiny fraction of the original amount of
carbon-14 is still present, then by appropriate laboratory measurements the
proportion of theoriginal amount of carbon-14 that remains can be
accurately determined.  In other words, if $Q(t)$ is the amount of carbon-14 at
time $t$ and $Q_{0}$ is the original amount, then the ration $\frac {Q(t)}{Q_0}$
can be determined, at least if this quantity is not too small.  Present
measurement techniques permit the use of this method for time periods up to about
100,000 years, after which the amount of carbon-14 remaining is only about $4 X
10^{-6}$ of the original amount. \newline
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(a) Assuming that $Q$ satisfies the differential equation $Q' = -rQ$, determine the
decay constant $r$ for carbon-14. \newline
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(b) Find an expression for $Q(t)$ at any time $t$, if $Q(0) = Q_0$. \newline
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(c) Suppose that certain remains are discovered in which the current residual
amount of carbon-14 is 20% of the original amount.  Determine the age of these
remains. \newline


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{\bf 3.}  A tank with a capacity of 500 gal. originally contains 200 gal. of water
with 100 lb. of salt in solution.  Water containing 1 lb. of salt per gallon is
entering at a rate of 3 gal/min, and the mixture is allowed to flow out of the
tank at a rate of 2 gal/min.  Find the amount of salt in the tank at any time
prior to the instant when the solution begins to overflow.  Find the concentration
(in pounds per gallong) of salt in the tank when it is on the point of
overflowing.  Compare this concentration with the theoretical limiting
concentration if the tank had infinite capacity.


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