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**Unformatted text preview: **CHAPTER 1: FIRST-ORDER DIFFERENTIAL EQUATIONS AND THEIR APPLICATION
19
b) Solve the differential equation and deter-
mine the amount in the account after 10 yr.
30. An amount of $10,000 is invested at 12% an-
c) How much more would the account have
nual interest compounded daily. An additional
after 10 yr if the full amount earned
investment of $B is made daily. What should
interest?
B be in order for the investment account to be
$100,000 after 10 yr?
28. An amount of $10,000 is deposited in a bank
daily.
that pays 9% annual interest compounded
31. An amount of $100 is deposited in a foreign
bank that pays 20% annual interest com-
pounded daily. Each day you make a transac-
a) If you withdraw $10 a day, how much money
tion of amount f() dollars. Part of the year
do you have after 3 yr?
you are able to deposit money [f(t) > 0],
b) How much can you withdraw each day
and part of the year you make withdrawals
if the account is to be depleted in exactly
[f(1) < 0]. If / is in years, these transactions
10 yr?
occur in the following cyclical yearly pattern:
29. A amount of $1000 is deposited in a bank that
pays 8% annual interest compounded daily. A
f ( 1 ) =
400
365 cos (2+1) $/day
deposit of B dollars is made daily.
= 400 cos (27rt) $/year.
a) What should B be in order to have $10,000
after 5 yr?
a) Find the amount of money in the account
b) Determine the function B(x) that gives the
as a function of t.
daily deposit needed to have $10,000 after x
years. [ B(5) is computed in part (a).]
b) Graph your answer to (a) for 10 yr (a sketch
will do if no computer is available).
1.9
Mixture Problems
In this section, a quantity Q(), such as the amount of some pollutant in a
water tank, varies with time. Further amounts of this quantity are being
added. The addition will be called inflow. Simultaneously, some of this
quantity is being lost. The quantity lost will be called the outflow. In the case
of a water tank, the loss could be due to evaporation, overflow, an open valve,
or all three. In all cases we assume that the tank is very well mixed (through
fast stirring if necessary), so that the concentration of the pollutant will be
assumed to be the same throughout all portions of the tank. The beginning
idea for analyzing these problems is the fundamental physical principle of
conservation of the quantity Q:
Rate of change
inflow rate
outflow rate
of Q
dt.
of Q
of Q
(1)
where we have noted that dQ/di, the time derivative of Q, is the rate of
change of Q with respect to time t....

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- Fall '10
- capretta