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1.8 GROWTH AND DECAY PROBLEMS
65
contain only 80% of the original amount of
radioactive material.
room is w. a constant. Find the desired
a) What is the half-life of this isotope?
room temperature as a function of w and
graph it.
b) How many additional years will it take until
only 15%% of the original amount is left?
23. An object at 100"C is to be placed in a 40'C
room. What should the constant of proportion-
20. At the time an item was produced, 0.01 of the
ality be in Eq. (25) in order that the object be
carbon it contained was carbon-14, a radioiso-
at 60"C after 10 min?
tope with a half-life of about 5745 yr.
24. The air in a room is cooling. At time / (in hrs)
a) You examine the item and discover that
the air temperature is Co(t) = 70 + 20e-/
only 0.0001 of the carbon is carbon-14. How
An object is placed in the room at time r = 0.
old is the object? (This process of determin-
The object is initially at 50C and changes
ing the age of an object from the amount of
temperature according to Eq. (25) with k = 1.
carbon-14 it contains is known as carbon-14
a) Find the temperature, T(r), of the object
dating.)
for 0 s t s 5.
b) Derive a formula that gives the age A of
b) Graph both Co and T on the same axes.
the object in terms of the fraction of carbon
that is carbon-14 at the present time, T.
25. An instrument at an initial temperature of
40'C is placed in a room whose temperature
21. The temperature of an engine at the time it is
is 20'C. For the next 5 h the room tempera-
shut off is 200"C. The surrounding air temper-
ture Qo(f) gradually rises and is given by
ature is 30"C. After 10 min have elapsed, the
Qo(f) = 20 + 10t, t in hours.
surface temperature of the engine is 180"C.
a) Give the form the cooling law (Eq. 25)
a) How long will it take for the surface tem-
takes for the instrument.
perature of the engine to cool to 40"C?
b) From prior experience, you know that your
b) For a given temperature 7 between 200"C
instrument cools according to Eq. (25) with
and 30'C, let (7) be the time it takes to
k = 1 if t is measured in hours. If T() is
cool the engine from 200'C to T. [For ex-
the surface temperature at time t, solve the
ample, /(200) = 0 and 1(40) is the answer to
equation in part (a) for T(1).
part (a).] Find the formula for {(7) in terms
c) Graph Qo() and T() on the same axes for
of T and graph the function. (The ambient
Osis5.
temperature is still 30.C.)
Many banks compound daily. Continuous com-
22. Earlier experiments have shown that a certain
pounding gives answers that are very close to those
component cools in air according to the cool-
given by daily compounding. In Exercises 26
ing law (Eq. 25) with constant of proportional-
through 31, approximate the effect of daily com-
ity 0.2. At the end of the first processing stage,
pounding by assuming continuous compounding.
the temperature of the component is 120"C.
26. You have $10,000 and intend to invest it for 5
The component remains for 10 min in a large
yr in a bank that offers continuous compound-
room and then enters the next processing stage.
ing. If you want to have $15,000 in your ac-
At that time the surface temperature is sup-
count at the end of these 5 yr, what annual
posed to be 60"C.
interest rate do you have to get?
a) What must the room temperature be for
27. You invest $2000 in an account paying 6% a
the desired cooling to take place?
year, compounded daily, on the amount in
b) Suppose that the entrance and exit temper-
excess of $500.
atures are still set at 120'C and 60"C, re-
a) Express this as a differential equation by
spectively, but the length of the wait in the
assuming continuous compounding....

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- Fall '10
- capretta