\begin{homework}\
\label{homework:18}

\medskip
\exercise
In Example 18.2, show that the solution given satisfies the initial
condition $y(0)=y_0$ and also the differential equation.  You will have
to use the trig formulas for $\sin(A\pm B)$, $\cos(A\pm B)$, and also
the triangle relating $\alpha$, $k$, and $\theta$.

\smallskip
\exercise
In Example 18.2, 

(a) find a formula for the initial temperature which would cause the
building's temperature function $y(t)$ immediately to equal the steady
state function;

(b) explain what happens to the phase shift (time delay) and the
oscillation amplitude of the steady state function if the building is very
well-insulated ($k$ is near 0) and if the building is very poorly insulated
($k$ is large); 

(c) find an approximate formula in terms of $k$ for the time delay in the
steady state function if the building is very well-insulated and if it is
very poorly insulated (note: the units of $k$ are time${}^{-1}$); (d) if
the outside temperature oscillates between $70^\circ$F and $90^\circ$F,
reaching $90^\circ$F at 3 p.m., while the inside temperature oscillates
between $75^\circ$F and $85^\circ$F, determine when the inside of the
building is hottest, and draw a graph of both the outside and inside steady
state temperature functions.

\smallskip
\exercise
In Problem 18(b) of the first problem set on differential equations,
suppose that the immigration $I$ is only 2 million people/year at time
$t=0$, but that it increases steadily each year by 250000 people/year.  As
before, suppose that the country's population is 300 million at time $t=0$,
and the death rate exceeds the birth rate by 1\%.  
Write a differential equation for the population function.  

\smallskip
\exercise
Suppose that a cup of coffee at temperature $y_0$ is brought into a room
whose temperature is $b$ at time $t=0$ but is increasing at the rate $a$ (in
${}^\circ$F/min).  Let $k$ denote the constant of proportionality in Newton's
law.  Find a differential equation for the temperature function $y(t)$.  

\smallskip
\exercise
In Problem 6 of the last problem set, suppose that, instead of fresh
water, water with 2 gm/liter of salt is entering the vat.  As before, the
vat contains 100 liters with $y_0$ grams of salt at time $t=0$, water comes
into the vat at $2+0.5t$ liters/min and leaves the vat at $2$ liters/min.
Give the differential equation satisfied by $y(t)$.

\smallskip
\exercise
Suppose that a vat contains $V$ liters of water and $y(t)$ grams of
salt.  Water is coming into the vat at constant rate $r$ liters/min and
leaving at the same rate.  The concentration of salt at time $t=0$ is $k$
gm/liters, but this concentration is increasing steadily at $\alpha$
gm/liter each minute.  Write a differential equation for $y(t)$.

\smallskip
\exercise
The normal mean temperature in Fairbanks, Alaska is closely approximated
by the sinusoidal function $$37\sin\left(\frac{2\pi}{ 365}(t-101)
\right)+25,$$
where $t$ is the number of days from the beginning of the year.  This
formula gives the expected average daily temperature during the $t$-th day
(obtained by averaging the temperature on the $t$-th day over a 30-year
period).\footnote{This formula for the temperature in Fairbanks is
taken from an article by B.~M.~Lando and C.~A.~Lando in the September 1977
issue of {\it The Mathematics Teacher}.}

Suppose that on April 11 you sell a building in Fairbanks (so take April
11, rather than the beginning of the year, to be $t=0$).  As part of the
purchase agreement, you are required to pay for electricity to keep the
building heated to $65^\circ$F for the first 4 years.  The cost of heating
the building is known to be \$0.50 for each degree-day below 65${}^\circ$,
and this cost is not expected to change during the 4-year period.  That is,
at any time $t$ the rate at which the heating bill accumulates is equal to
\$0.50 times $(65-\text{current~temperature})$.  To cover your obligation
to pay for heating you leave an amount $y_0$ of money with the electric
company.  The electric company agrees to pay interest $k$ (compounded
continuously) on the money, and also each day deduct the heating cost for
that day.  (We make the simplifying assumption that this is a continuous
process, rather than something that happens once a day.  In the same way,
when writing the above formula for the temperature function in Fairbanks we
suppose that $t$ is a continuous variable.)

(a) Write a differential equation for the amount of your money the electric
company has at time $t$.

(b) If $k=0$, i.e., if the electric company pays no interest on the money
you deposited with them, how much should you leave with them to cover the 4
years' heating cost?

(c) Now suppose that $k$ is $9\frac{1}{ 8}\%$ annual interest (i.e.,
$k=0.09125~\text{yr}^{-1}$ --- but be careful to use consistent time
units).  Find a formula for $y(t)$ in terms of $t$ and the constant $y_0$.

(d) In the situation of part (c), how much money should you leave with the
electric company to exactly cover the heating cost for 4 years?

\smallskip
\exercise
At time $t=0$ a vat contains 100 liters of water with $y_0$ grams of
salt.  At that time you start letting water flow out of the vat at a rate
$r(t)$ that depends on time: the rate starts at 12 liters/min at time
$t=0$, and it decreases at $a$ liters/min${}^2$ (until it reaches 0, at
which point the experiment stops).  There is no water coming into the vat,
but someone is sprinkling salt into the water at the rate $l$ gm/min.  

(a) Write formulas for $r(t)$ and the volume $V(t)$.  

(b) Find the differential equation satisfied by the amount $y(t)$ of salt
at time $t$.  (The constants $a$ and $l$ should appear in your differential
equation.)  

\smallskip

\exercise
At time $t=0$ we have $x_0$ grams of radioactive substance A and $y_0$
grams of radioactive substance B.  The former has half-life $H_1$, and so
has radioactive decay constant $k_1=(\ln 2)/H_1$; the latter has a
different half-life $H_2$ and decay constant $k_2=(\ln 2)/H_2$.  The
substance A decays into the substance B, which then, in turn, decays into
some non-radioactive substance.  

(a) Find a formula for the amount $x(t)$ of A at time $t$.

(b) Find a differential equation satisfied by the amount $y(t)$ of $B$ at
time $t$.


\smallskip
\begin{rightindent}{3.25 truein}
\exercise
The diagram at the right shows two vats, the first containing $V$ liters of
water and the second containing $W$ liters of water.  Water with $k$
gm/liter of impurity is entering the first vat at constant rate $r$, water
is passing from the first vat to the second vat at the same rate, and water
is leaving the second vat also at rate $r$.  Let $x(t)$ be the amount of
impurity in the first vat at time $t$, and let $y(t)$ be the amount in the
second vat at time $t$.  
\end{rightindent}
\hfill
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(a) Find a formula for $x(t)$.  

(b) Find a differential equation satisfied by $y(t)$.

\end{homework}
