\begin{homework}\
\label{homework:16}
\medskip

\exercise
Suppose you buy an \$800 stereo system, taking 3 years to pay at 16\%
interest with no money down.  What are your monthly payments, and how much
will you end up paying for the stereo in all?

\smallskip
\exercise
Suppose you want to buy a \$20000 boat but have only \$5000 for a
downpayment.  You can get a 5-year loan for the rest at 18\%.  

(a) What are
your monthly payments, and how much will you end up paying for the boat?

(b) If you can only afford \$300 monthly payments, how long will it take
you to pay off the loan?  

(c) Suppose you arrange a loan where you are
paying \$300 per month as in part (b).  How much of the \$3600 that you pay
in the first year is tax-deductible?  How much of the \$3600 that you pay
in the fifth year is tax-deductible?

\smallskip
\exercise
Suppose that a mortgage company will give you a 30-year mortgage at 12\%
interest, but they consider you an acceptable credit risk only if you make
a 20\% downpayment and if your monthly payments do not exceed one fourth of
your \$1600 monthly salary.  What is the most expensive house you can buy,
and how much cash on hand do you need to buy it?

\smallskip
\exercise
You buy a house for \$90000 and make a \$20000 downpayment, using a
30-year mortgage for the rest. 

(a) Suppose that between the time you agree
to buy the house and the time the arrangements are finalized, the interest
rate jumps from 11\% to 12\%.  What increase in the price of the house
would have been equivalent to the increase in interest rate (i.e., would
have led to the same increase in your monthly payments, assuming your
downpayment remains the same)? 

(b) Suppose that the interest rate is 12\%.
How much more would you have to pay each month if you want to pay off the
mortgage in 25 years rather than 30 years?  

(c) What is the shortest term
mortgage for which your monthly payments do not exceed \$750?

\smallskip
\exercise
An item is bought using a loan of $y_0$ dollars, which is
paid back at the steady rate of $a$ dollars/year.  The instantaneous
interest rate is 10\% (assume continuous compounding), and the term of the
loan is 20 years.  At what instant $t$ is exactly half of the money going
for interest (i.e., half of the $a$ dollars per year being paid is going
for interest at that instant) and half for principal?  

\smallskip
\exercise
Suppose that on your 16th birthday a wealthy distant relative dies and
leaves you \$30000 in a trust fund that works as follows.  The money in the
fund always receives 8\% interest compounded continuously.  You receive no
money until you turn 21.  Then for the next 30 years you receive a fixed
amount each month, which is determined in such a way that the fund is
precisely used up on your 51st birthday.  (Make the simplifying assumption
that you receive money continuously throughout the month rather than in a
lump payment once each month.)  

(a) How much money is in the trust fund
when you turn 21?  

(b) How much will you receive each month between your
21st and 51st birthdays?  

(c) If you decide to take only \$300 per month
and leave the rest in the fund, how long will the trust fund last?  

(d) How
much can be withdrawn per month if you want the trust fund to last forever?

 
\smallskip 
\exercise
The table below is based on data published in the June
15, 1981 issue of The Seattle Times. It shows the monthly payments and total
cost for a 42-month \$6000 loan for a new car, depending on the interest rate
demanded by various lending institutions.

\begin{centering}
\sf\small
\begin{tabular}{||c|c|c|c||} \hline
\multicolumn{4}{c}{{\large Basic new car loan \$6,000, 42 months}}
\\ \hline
LENDER & INTEREST RATE & MONTHLY PAYMENT & TOTAL COST
\\ \hline 
Boeing Credit Union         & 13.75\%   & \$180.78   & \$7,792.76 
\\ \hline
Auto Dealer No.1            & 16\%      & \$187.50   & \$7,875.00
\\ \hline
First Interstate Bank       & 18\%      & \$193.59   & \$8,130.78
\\ \hline
Auto Dealer No.2            & 18.25\%   & \$194.35   & \$8,162.72
\\ \hline
Rainer National Bank        & 19.25\%& \$197.45& \$8,292.90
\\ \hline
Seattle-First National Bank &20.5\%     & \$201.35   & \$8,456.70
\end{tabular}\\
\end{centering}

\smallskip

\noindent
Use the procedure in this section to compute the monthly payments in each
of the six cases, and see if your answers agree with the values given.  They
will not be exactly the same, because of our simplifying assumptions.
However, you should not be more than about a dollar off.

\smallskip
\exercise
You make a pudding for a dinner party, and put it in the refrigerator at
6 p.m. ($t=0$).  Your refrigerator maintains a constant temperature of
40${}^\circ$.  The pudding will gel when it cools to 45${}^\circ$.  When
you put the pudding in the refrigerator you measure its temperature to be
190${}^\circ$, and when the first guest arrives at 7 p.m. you measure it
again and get a temperature reading of 100${}^\circ$.  When is the earliest
you can serve dessert?

\smallskip
\exercise
Water at 55${}^\circ$ is pumped into a swimming pool on a 90${}^\circ$
summer day.  After 2 hours the temperature of the water is 60${}^\circ$.
In how many hours (assuming the outside temperature remains 90${}^\circ$)
will the water reach a comfortable swimming temperature of 70${}^\circ$?

\smallskip
\exercise
A detective discovers a murder victim in a room at 8:00 a.m.~one
morning.  The temperature of the body is 80.0${}^\circ$F.  Two hours later
(at 10:00 a.m.) the temperature of the body is 75.0${}^\circ$F.  Assuming
that the room is kept at the constant temperature 70.0${}^\circ$F,
determine the time at which the murder was committed.  (Assume that the
murder victim had a normal temperature of 98.6${}^\circ$F at the time of
death.)

\smallskip
\exercise
The detective in Problem 10 discovers another murder victim at noon the
next day in another 70.0${}^\circ$F room, and immediately has the body
transported to a morgue, where the temperature is kept at 50.0${}^\circ$F.
She measures the body's temperature right before it is taken from the room,
and again at 4 p.m., after it has been 4 hours in the morgue.  The former
temperature is 85.0${}^\circ$F, and the latter temperature is
65.0${}^\circ$F.  When was the murder committed?  (Assume that the same
constant of proportionality in Newton's law applies in the room and in the
morgue.)

\smallskip
\exercise
Suppose you drop an object from a great height, so that it reaches very
nearly terminal velocity by time it hits the ground.  By measuring the
impact you determine that this terminal velocity is 49 m/sec. 

(a) Find $k$.

(b) Graph the velocity function for the following values of $v_0$: 0
(dropped from rest), 20 m/sec downward, 100 m/sec downward, 49 m/sec
downward, 20 m/sec upward.  

(c) At what velocity must the object be thrown
upward if you want it to reach its peak height after 3 sec?  

(d) If $v_0$
is 20 m/sec upward, how long does the object take to reach its peak height?

\smallskip
\exercise
In Example 16.4, find formulas for the distance $s(t)$ fallen from the
plane at time $t$ for $0\le t\le 10$ and for $t>10$.  Notice that the
exponential decay term makes a negligible contribution when $t$ is greater
than about 15 sec.  If the plane was at an altitude of 500 meters when the
parachutist jumped, when does she reach the ground?
 
\smallskip
\exercise
A dense heavy package is dropped from a plane.  After 7 seconds the
package releases a parachute that starts to slow its speed.  Assume that
the air resistance is negligible during the first 7 seconds, and that wind
tunnel tests show that with the parachute open the air resistance balances
gravity at a speed of $19.6$ m/sec.  Find formulas for the downward
velocity $v(t)$ for $0\le t\le 15$, and graph $v(t)$.

\smallskip
\exercise
Water is flowing at 3 m${}^3$/min in one end and at the same rate out
the other end of a vat containing 100 m${}^3$ of water.  Let $y(t)$
be the amount (in grams) of impurity in the vat of water at time $t$ min.
The water flowing in contains 0.1 gm of impurity per cubic meter.  Find
$y(t)$ if the initial concentration of impurity in the container is 

(a) 0.05 gm/m${}^3$, (b) 0.20 gm/m${}^3$, (c) 0.10 gm/m${}^3$.  

\noindent
Graph $y(t)$ in each case (on the same $xy$-plane).

 
\exercise
A tank originally contains 100 gal of fresh water.  Water with 1/2
lb/gal of salt is poured into the tank at a rate of 2 gal/min, and the
well-stirred mixture is allowed to leave at the same rate.  After 10
minutes the process is stopped, and fresh water is poured into the tank at
a rate of 2 gal/min, with the mixture again leaving at the same rate.  Find
the amount of salt in the tank at the end of 20 minutes.

\smallskip
\exercise
Suppose that a room containing 1200 cubic feet of air is originally
free of carbon monoxide.  Beginning at time $t=0$ cigarette smoke
containing 4 percent carbon monoxide is introduced into the room at a rate
of $0.1$ ft${}^3$/min, and the well-circulated mixture is allowed to leave
the room at the same rate.  

(a) Find an expression for the amount $y(t)$ of
carbon monoxide (in cubic feet) in the room at time $t$.  

(b) Extended
exposure to a carbon monoxide concentration as low as 0.012\% is harmful to
the human body.  Find the time at which this concentration is reached.

\smallskip
\exercise
Consider a lake of constant volume $V$ which at time $t$ contains an
amount $y(t)$ of pollutant, evenly distributed throughout the lake with a
concentration $c(t)$, where $c(t)=y(t)/V$.  Assume that water containing a
concentration $k$ of pollutant (measured in tons/km${}^3$) enters the lake
at a rate $r$ (measured in km${}^3$/yr), and that water leaves the lake at
the same rate.  Suppose that pollutants are also added directly to the lake
at a constant rate $P$ (measured in tons/yr).  Note that the given
assumptions neglect a number of factors which may be important in some
cases: the water added or lost by precipitation, absorption, and
evaporation; the stratifying effect of temperature differences in a deep
lake; the tendency of irregularities in the coastline to produce sheltered
bays; and the fact that pollutants are not deposited evenly throughout the
lake but rather at isolated points around its periphery.  However, we shall
neglect these factors in what follows.  Thus, according to our assumptions,
the rate of change of $y(t)$ is the sum of three contributions: $rk$
tons/yr in the incoming water, $P$ tons/yr entering additionally, and {\it
minus} $r$ times $y/V$ leaving in the outgoing water.

(a) Write a differential equation for $y$.

(b) Solve the differential equation to find an expression for $y(t)$.  
The constants  $r$, $k$, $P$, $V$, and $y_0$ should appear in
your expression.

(c) What is the limiting value of $y(t)$ as $t\longrightarrow\infty$?

(d) If the addition of pollutants to the lake is terminated (i.e., $k=0$
and $P=0$ for $t>0$), determine the time interval $T$ which must elapse
before the concentration of pollutants is reduced to 10\% of its original
value.

(e) The table below contains data for several of the Great Lakes.  Use the
data to determine the time $T$ necessary to reduce the contamination of
each of these lakes to 10\% of the original value.

Lake Superior: $V=12200$ km${}^3$, $r=65.2$ km${}^3$/yr

Lake Michigan: $V=4900$ km${}^3$, $r=158$ km${}^3$/yr

Lake Erie: $V=460$ km${}^3$, $r=175$ km${}^3$/yr

Lake Ontario: $V=1600$ km${}^3$, $r=209$ km${}^3$/yr

\smallskip
\exercise
Suppose that at time $t=0$ a country's population is 300 million, and
its death rate exceeds its birth rate by k.  Normally, that would lead to
exponential decay of the population given by the formula $y(t)=y_0e^{-kt}$.
However, there is also a constant immigration of $I$ million people per
year.  

(a) Write a differential equation for the population function.  

(b) If $k=1\%$ and $I=4$ million people per year, find the population in 50
years, and also find the limiting value of the population.

\smallskip
\exercise
The population of salmon in the Pacific Northwest, if not harvested,
would grow by 10\% per year (this is effective yearly rate, {\bf not}
instantaneous rate).  Suppose that the current salmon population is 3
million, and that the salmon are being harvested at a constant rate of $R$
per year.  At what rate $R$ should the salmon be harvested in order that
the salmon population be 4 million in 4 years from now?  What would happen
if $R$ were 300000 salmon per year?

 
\end{homework}

 
%---------------------------------------------------------------
