EXAMPLE 18
According to Newton’s law of cooling, a hot object cools at a rate proportional to the difference
between its own temperature and that of its environment. If a roast at room temperature 68°F is
put into a 20°F freezer, and if, after 2 hours, the temperature of the roast is 40°F:
(a) What is its temperature after 5 hours?
(b) How long will it take for the temperature of the roast to fall to 21°F?
SOLUTIONS: This is an example of Case II (b) (the temperature is decreasing toward the
limiting temperature 20°F).
(a) If R(t) is the temperature of the roast at time t, then
(b) Equation (*) in part (a) gives the roast’s temperature at time t. We must find t when R = 21:EXAMPLE 19
Advertisers generally assume that the rate at which people hear about a product is proportional
to the number of people who have not yet heard about it. Suppose that the size of a community is
15,000, that to begin with no one has heard about a product, but that after 6 days 1500 people
know about it. How long will it take for 2700 people to have heard of it?
SOLUTION: Let N(t) be the number of people aware of the product at time t. Then we are
given that
N ′(t) = k[15,000 − N(t)],
which is Case IIa. The solution of this d.e. is
N(t) = 15,000 − ce−kt.
Since N(0) = 0, c = 15,000 and
N(t) = 15,000(1 − e−kt ).
Since 1500 people know of the product after 6 days, we have