The following problems deal with the function payment that we derived
earlier in this lesson for computing time payments on a loan financed
at some annual interest. For the sake of argument, let's assume the loan is
for $100,000 and that the annual interest rate is 7%.
(1) What should your monthly payment be if you want to pay off the loan in 30
years?
(2) What is the total amount of interest that you pay over the life of the loan
if you use the monthly payment calculated in part (1)?
(3) How long will it take to pay off the loan if you double the monthly payment
that you calculated in part (1)? How much interest will you save this way?
(4) Prepare a graph showing how the loan balance declines over 30 years if you
choose either the monthly payment calculated in part (1) or used in part (3).
Put both curves on the same plot. Use what you've learned about plotting to
label the plot sensibly.
The next two problems deal with a generalization of the loan problem.
(5) Derive a function called ``principal'' that takes four arguments: a
month number i, a starting loan amount P, an interest rate R, and a
monthly payment M. principal[i, P, R, M] should give the outstanding
balance of a $P loan after i months, where the interest rate is R and
the monthly payment is M. You'll need to repeat many of the steps of the
original derivation, using symbols instead of numbers.
(6) If you can get a loan at 6.5% interest, and you want to make monthly
payments of $800, what is the largest loan that you can pay off in 30 years?
We'll now return to the destroyer example by finishing up the calculation to
determine the power required for the destroyer to traverse the course that
we specified. Here are the steps.
(7) Derive a function ``acceleration'' that takes time as an argument and
returns the x and y components of the destroyer's acceleration. It will give
acceleration in meters/sec/sec. Acceleration is the derivative of velocity in
the same way that velocity is the derivative of position.
(8) Derive a function ``force'' that takes time as an argument and returns
the x and y components of the force that must be placed upon the destroyer (by
its engines and by the water's drag) in order to produce the desired
acceleration. To obtain the force function, you must multiply each component
of the acceleration by the mass of the destroyer, much as you took the
derivative of each component of the velocity. The mass of the destroyer is
5000 tons, which you should convert to kilograms so that your resulting
function will express force in newtons. (There are 2000 pounds in a ton, and
there are 2.2 pounds in a kilogram.)
(9) Once you have a force function, the following ``power'' function will
compute the power required to move the destroyer along the prescribed
trajectory.
power[t_]:=
Abs[(force[t] + velocity[t]*1100*speed[t]).velocity[t]]
(The number 1100. that appears in the formula is the destroyer's drag
coefficient, which is a measure of how the drag of the water affects the
destroyer's motion.)
The ``power'' function will return the power requirements in units of watts,
and will be a scalar function of t. (It'll also be quite a forbidding
looking function---don't let it scare you!) Plot the function to help find the
peak power required during the maneuver. You don't need to hand in the plot,
but be sure to explain how you used to plot to find the peak power requirement.
(If you've done everything right, you'll find that the power at time t=2 is
approximately .955*10^8 watts, and the power at time t=20 is approximately
.161*10^9 watts. It turns out that we're asking too much of our destroyer. A
modern destroyer can generate a peak power of around 10^7 watts.)
(10) How many 6-cylinder Toyota pickup engines would be required to produce the
peak power? (Each engine can produce 140 horsepower, and one horsepower equals
about 745.7 watts.)
Prepare a report giving the answers to these ten questions. Be sure that your
report includes the question, your answer, and relevant portions of a Mathematica
transcript that supports your answers. Please prepare a neat report---a
sloppy one won't impress anyone, least of all your TA.