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 200 degrees Fahrenheit when freshly poured, and 2.5 minutes later has cooled to 180 degrees in a room at 76 degrees, determine when the coffee reaches a temperature of 130 degrees. The coffee will reach a temperature of 130 degrees in minutes.

Answer 1

Answer:

Step-by-step explanation:

As per Newton law of cooling we have temperature of a cooling object at time t is given by

T(t) = $T_s + (T_0-T_s) e^{-kt}$

where $T_s = 76:   T_0 = 200$

When t=2.5, we have

$T(2.5) = 76+(100-76)e^{-2.5k} =180\\e^{-2.5k} =\frac{104}{24} =4.333\\k=\frac{ln4.333}{-2.5} =-0.5865$

Hence equation is

$T(t) = 76+24e^{-0.5865t}$

Using this we find that

$130=76+24e^{-0.5862t} \\e^{-0.5862t}=54\\t=1.59559$

i.e. At 1.6 seconds the coffee will reach a temperature of 130 degrees