
\begin{homework}\
\label{homework:11}

\medskip
\exercise
Suppose that interest on money in the bank accumulates at an annual rate
of 7\% compounded continuously.  How much should be invested today so that
20 years from now it will be worth \$20000?
B

\smallskip
\exercise
Suppose that a sum of money draws interest compounded continuously.  How
long will it take to double if the annual rate is (a) 6\%?, (b) 10\%?, (c)
18\%?  How long will it take to double if the effective annual rate is (a)
6\%?, (b) 10\%?, (c) 18\%?

\smallskip
\exercise
In an experiment to study the growth of bacteria, a medical student
measured 5000 bacteria at time 0 and 8000 at time 10 minutes.  Assuming
that the number of bacteria grows exponentially, how many bacteria will be
present after 30 minutes?

\smallskip
\exercise
Suppose that the population of a certain species grows at the
instantaneous rate of 2\% per year.  If the present population is 1000000,
what will the population be in 1 year?  In 20 years?  How long will the
population take to double?

\smallskip
\exercise
Assume that the population of the Earth changes at a rate proportional
to its current population.  Assume further that in A.D.~1650 its population
was 250 million, and in A.D.~1950 its population was 2.5 billion.  Find an
expression giving the population of the Earth at any time.  If the greatest
population the Earth can support is 25 billion, when will this limit be
reached?

\smallskip
\exercise
If you hear on the news that	prices are rising exponentially at the
effective yearly rate of 12\%, what is the instantaneous annual rate of
inflation?  How long does it take prices to double at this rate?

\smallskip
\exercise
Suppose that the average yearly cost of college in the U.S. (tuition and
expenses) is increasing at the instantaneous rate of 10\% per year.  By
what factor will the cost of college increase for a child born now in the
18 years before she goes to college?

\smallskip
\exercise
Suppose that bank interest is compounded continuously at 6\%, and
inflation on food has instantaneous rate 13\%.  (a) If you put \$10000 in
the bank, how much will you have 100 years later?  (b) If it now costs
\$100/week to feed an average family, write a formula for weekly food cost
$t$ years later.  (c) What is the {\bf total} cost of feeding a family over
the next 10 years?  (d) What will the weekly food bill be in 100 years?
(e) If your great grandchild takes your money in part (a) out of the bank
at that time, how long will it feed his or her family?

\smallskip
\exercise
The half-life of Carbon 14 is 5568 years.  Archeologists find an ancient
table made from a tree which had $y_0$ grams of Carbon 14 when it was cut.
Now the wood has only $y_1$ grams of Carbon 14.  Express the age of the
table in terms of $y_0$ and $y_1$.

\smallskip
\exercise
A certain radioactive material has a half-life of 2 hours.  Find the
time needed for a given amount of this material to decay to one-tenth of
its original mass.

\smallskip
\exercise
A certain radioactive material is known to decay at a rate proportional
to the amount present.  A block of this material originally having a mass
of 100 grams is observed after 20 years to have a mass of only 80 grams.
Find an expression for the mass of the material at any time.  Also find the
half-life of the material.

\smallskip
\exercise
Suppose that the average yearly real income (i.e., in
inflation-adjusted dollars, say 1995-dollars) of a U.S.~worker is now
decreasing at the instantaneous rate of 1.5\% per year, and that this trend
continues indefinitely.  If the current average salary is \$18000, what
will an average worker's yearly income (in 1995-dollars) be in 30 years?

\smallskip
\exercise
You are flying in a plane at constant altitude $y_0$, constant
speed $v_0$ and constant direction.  You want to drop a container of relief
supplies to a target that is straight ahead under your flight path.  You will
release the container at the instant when your line of sight to the
target makes an angle $\alpha$ below the horizontal.  The purpose of this
problem is to determine $\alpha$.  Let $v(t)$ denote the $\underline{\rm
horizontal}$ component of the velocity of the container.  Assume that air
resistance causes a horizontal component of deceleration that is
proportional to velocity, with
proportionality constant $k$.  Air resistance is the only force acting
to change the velocity (since gravity does not affect horizontal motion).

(a) Write a differential equation for $v(t)$, and find a formula for
$v(t)$.

(b) Find a formula for the horizontal distance traveled by the container
between time 0 and time $t$.

(c) Suppose that after enough time has elapsed so that the exponential
decay term in part (c) is negligible, a parachute opens and the container
glides straight down until it hits the ground.  Find $\alpha$. 

\end{homework}
 
%--------------------------------------------------
 
