150

CHAPTER 2

SECTION 2.4

Linearity and Nonlinearity

Linear Models: Mixing and Cooling

−k t +2 or 48.6e ( 1 ) = 10 . Dividing the second equation by the first equation gives the 1 ln 2 relationship e −2 k = from which k = . Using this value for k the equation for T ( t

2

) 1

2

gives 70 = 50 + 48.6e −t 1 ln 2 2 from which we find t1 ≈ 2.6 hours. Thus, the person was killed approximately 2 hours and 36 minutes before 8 P.M., or at 5:24 P.M. (b)

Following exactly the same steps as in part (a) but with T0 = 98.2° F, the sequence of equations is T(t1) = 70 = 50 + 48.2e − k (t1 ) ⇒ 48.2e − kt1 = 20. T(t1 + 2) = 60 = 50 + 48.2e− k (t1 + 2) ⇒ 48.2e − k (t1 + 2) = 10. Dividing the second equation by the first still gives the relationship e−2k = so k =

1 , 2

ln 2 . 2

Now we have T(t1) = 70 = 50 + 48.2e −t1 ln 2 / 2 which gives t1 ≈ 2.54 hours, or 2 hours and 32 minutes. This estimates the time of the murder at 5.28 PM, only 4 minutes earlier than calculated in part (a). 

A Real Mystery

22.

T ( t ) = T0 e− kt + M 1 − e − kt

(

)

While the can is in the refrigerator T0 = 70 and M = 40 , yielding the equation T ( t ) = 40 + 30e − kt . Measuring time in minutes, we have T (15 ) = 40 + 30e −15k = 60 , ⎛ 1 ⎞ ⎛ 2⎞ which gives k = − ⎜ ⎟ ln ⎜ ⎟ ≈ 0.027 . Letting t1 denote the time the can was removed from the ⎝ 15 ⎠ ⎝ 3 ⎠ refrigerator, we know that the temperature at that time is given by T ( t1 ) = 40 + 30e

− kt1

,

150

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...

Solutions manual for differential equations and linear algebra 2nd edition by farlow ibsn 9780134689

Solutions Manual for Differential Equations and Linear Algebra 2nd Edition by Farlow IBSN 9780134689548 Full clear download( no error format...