Question

Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of 190190 degrees Fahrenheit when freshly poured, and 11 minutes later has cooled to 172172 degrees in a room at 6060 degrees, determine when the coffee reaches a temperature of 122122 degrees.

Answer 1

4.9 minutes

Step-by-step explanation:

Given; T(t) = Ce^-kt + Ts

Now;

T(t) = 190 degrees Fahrenheit

Ts = 60 degrees

To obtain C;

190 = Ce^0 + 60

190 - 60 = C

C = 130

Hence, to find k when t=11

172 = 130 e^-11k + 60

172 -60/130 = e^-k

e^-k = 0.86

ln(e^-k) = ln( 0.86)

-k = -0.15

k = 0.15

Hence at 122 degrees, t is;

T(t) = Ce^-kt + Ts

122 = 130e^-0.15t + 60

122 - 60/130 = e^-0.15t

0.477 = e^-0.15t

ln (e^-0.15t) = ln (0.477)

-0.15t = -0.74

t = 0.74/0.15

t = 4.9 minutes