Page 75 - Schaum's Outline of Differential Equations

58 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 7

We could solve (5) for N, but this is not necessary. We seek a value of t when N= 250, one-half the population.
Substituting N = 250 into (5) and solving for t, we obtain

or t = 0.009191k time units. Without additional information, we cannot obtain a numerical value for the constant of
proportionality k or be more definitive about t.

7.8. A metal bar at a temperature of 100° F is placed in a room at a constant temperature of 0°F. If after
20 minutes the temperature of the bar is 50° F, find (a) the time it will take the bar to reach a temperature
of 25° F and (b) the temperature of the bar after 10 minutes.
Use Eq. (7.2) with T m = 0; the medium here is the room which is being held at a constant temperature of 0° F.
Thus we have

whose solution is

Since T= 100 at t = 0 (the temperature of the bar is initially 100° F), it follows from (1) that 100 = ce- k(0) or 100 = c.
Substituting this value into (1), we obtain

At t= 20, we are given that T= 50; hence, from (2),

from which

Substituting this value into (2), we obtain the temperature of the bar at any time t as

(a) We require t when T = 25. Substituting T=25 into (3), we have

Solving, we find that t = 39.6 min.

(b) We require T when t = 10. Substituting t= 10 into (3) and then solving for T, we find that

It should be noted that since Newton's law is valid only for small temperature differences, the above
calculations represent only a first approximation to the physical situation.

7.9. A body at a temperature of 50° F is placed outdoors where the temperature is 100° F. If after 5 minutes
the temperature of the body is 60° F, find (a) how long it will take the body to reach a temperature of
75° F and (b) the temperature of the body after 20 minutes.
Using (7.2) with T m = 100 (the surrounding medium is the outside air), we have