Page 175 - How To Solve Word Problems In Calculus

EXAMPLE 9
According to Newton’s law of cooling, the temperature of an
object changes at a rate proportional to the difference in tem-
perature between the object and the outside medium. If an ob-
◦
ject whose temperature is 70 Fahrenheit is placed in a medium
◦
whose temperature is 20 and is found to be 40 after 3 min-
◦
utes, what will its temperature be after 6 minutes?
Solution
If u(t) represents the temperature of the object at time t,
du
represents its rate of change. Newton’s law of cooling may
dt
du
be written = k(u − 20).
dt
1 du
= k
u − 20 dt
d
ln(u − 20) = k
dt

ln(u − 20) = kt + C

We can solve for C by observing that u(0) = 70

ln(70 − 20) = k · 0 + C

C = ln 50

Next we determine u(t).

ln(u − 20) = kt + ln 50

ln(u − 20) − ln 50 = kt

u − 20
ln = kt
50
u − 20 kt
= e
50
u(t) = 20 + 50e kt

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