Barrons AP Calculus

(a)

(b)

(a)

(b)

and

y   =   A   −   ce−kt.

Case    II(b)   can be  solved  similarly.

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:

What    is  its temperature after   5   hours?

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).

If  R(t)    is  the temperature of  the roast   at  time    t,  then

Since   R(0)    =   68°F,   we  have

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