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(d) Redo part (c) but place the Captain halfway to one side (that is,
at
x
=
37
.
5,
y
=
25 if the coordinates of the arena are 0
≤
x
≤
50
,
0
≤
y
≤
50).
(e) Redo the simulation with the Captain completely to one side, and
finally in a corner. What self-evident fact is reinforced for you?
2. Consider an account that has
M
dollars in it and pays monthly interest
J
.
Suppose beginning at a certain point an amount
S
is deposited monthly
and no withdrawals are made.
(a) Assumefirstthat
S
=
0.Usingthe
MortgagePayments
applicationin
Chapter 9 as a model, derive an equation relating
J
,
M
, the number
n
of months elapsed, and the total
T
in the account after
n
months.
Assume that the interest is credited on the last day of the month
and that the total
T
is computed on the last day after the interest
is credited.
(b) Now assume that
M
=
0, that
S
is deposited on the first day of the
month, that as before interest is credited on the last day of the
month, and that the total
T
is computed on the last day after the
interest is credited. Once again, using the mortgage application as
a model, derive an equation relating
J
,
S
, the number
n
of months
elapsed, and the total
T
in the account after
n
months.
(c) By combining the last two models derive an equation relating all of
M
,
S
,
J
,
n
, and
T
, now of course assuming there is an initial amount
in the account (
M
) as well as a monthly deposit (
S
).
(d) If the annual interest rate is 5%, and no monthly deposits are made,
how many years does it take to double your initial stash of money?
What if the annual interest rate is 10%?
(e) Inthisandthenextpart,thereisnoinitialstash.Assumeanannual
interest rate of 8%. How much do you have to deposit monthly to be
a millionaire in 35 years (a career)?
(f) If the interest rate remains as in (e) and you can only afford to
deposit $300 each month, how long do you have to work to retire a
millionaire?
(g) You hit the lottery and win $100,000. You have two choices: Take
the money, pay the taxes, and invest what's left; or receive $100,000/
240 monthly for 20 years, depositing what's left after taxes. Assume
a $100,000 windfall costs you $35,000 in federal and state taxes, but
that the smaller monthly payoff only causes a 20% tax liability. In
which way are you better off 20 years later? Assume a 5% annual
interest rate here.
(h) Bankspayroughly5%,thestockmarketreturns8%onaverageover
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