You place a cup of 205°F coffee on a table in a room that is 72°F, and 10 minutes later, it is 195°F. Approximately how long will it be before the coffee is 180°F?

1 Answer
May 20, 2017

Answer:

Approximately #27# minutes.

Explanation:

This is a Newton's Law of Cooling Problem.

Newton's Law of Cooling states that an object cools down by the formula #T(t) = T_s + (T_0 - T_s)e^(-kt)#, where #t_0# is the initial temperature of the liquid, #t_s# is the temperature of the surrounding environment, #t# is the number of minutes elapsed and #T(t)# is the temperature after #t# minutes elapsed. #k# is the constant, which will differ depending on the object.

We have to start by finding #k#.

#195 = 72 + (205 - 72)e^(-k(10))#

#123/133 = e^(-10k)#

#ln(123/133) = -10k#

#k = -1/10ln(123/133)#

So our formula is

#T(t) = 72 + 133e^(1/10ln(123/133)t)#

We're looking for the amount of time it takes for the coffee to cool to #180# degrees fahrenheit, so we write the equation:

#180 = 72 + 133e^(1/10ln(123/133)t)#

#108/133 = e^(1/10ln(123/133)t)#

#ln(108/133) = 1/10ln(123/133)t#

A good approximation gives

#t ~~ 26.64#

Therefore, it will take approximately #27# minutes for the coffee to cool to #180˚#F.

Hopefully this helps!