Exponential Growth/Decay

Question

A frozen steak initially has a temperature of 26°F. It is left to thaw in a room that has a temperature of 71°F.

After 9 ​minutes, the temperature of the steak has risen to 36°F. After how many minutes will the temperature of the steak be 50°​F? Use​ Newton's Law of​ Cooling to solve.

Answer 1

T(t) = Tenv + (T(0)-Tenv)e^-rt

50= 71 + (26-71)e^-rt

-21/-45 =e^-rt

ln(7/15) = -rt

t= -ln(7/15)/r

after 9 minutes it's temp has risen to 36, use this to solve for r, and plug that value into above to find t = time

when it reaches 50 degrees

36 = 71 +(26-71)e^-9r

-35/-45 =7/9= e^-9r

r =- [ln(7/9)]/9

t = 9ln(7/15)/ln(7/9)= about 27.2935 minutes

= about 27 minutes to reach 50 degrees

26 degrees t=0

36 degrees t=9

50 degrees t=27

or Newton's (negative) cooling

9 minutes=.15 hours

T(t)= 71 +(26-71)e^kt

T(3/20)= 36= 71-45e^(3k/20)

-35/-45 =7/9 = e^.15k

k= (ln7/9)/.15= about -1.675

50=71-45e^(ln(7/9)t/.15

-21/-45= e^ln(7/9)t/.15

ln(7/15)= ln(7/9)t/.15

t= .15ln(7/15)/ln(7/9)= about .455 hours= about 27.29 minutes