Newton\'s Law of Cooling relates the rate at which a body cools to the differenc
ID: 3024723 • Letter: N
Question
Newton's Law of Cooling relates the rate at which a body cools to the difference in temperature between the body and the environment into which it is introduced, The formula is
F(t)=To+Cb^t where F(t) is the temperature of body at time t. To is the constant temperature of the environmet into which the body is introduced, and C and b are constants. Use Newtons Law of Cooling to solve this problem.
A cup of coffee has cooled from 99 degrees to 60 degrees after 15 minutes in a room at 20 degrees. How long will it take to cool to 40 degrees?
Explanation / Answer
Hi, I am Waqar. Please find the solution below:
Since we are given with the expression F(t)=T0+Cbt and we are given with the environmental temperature T0 = 20 degrees. Also we are given with the temperate at time 0 and at time 15
=> F(0) = 99 & F(15) = 60
Now we will solve the expression F(t) for t = 0 & t = 15
=> F(0) = 20 + Cb0 = 99
=> 20 + C x 1 = 99 ( Anything raised power Zero is 1, i.e b0 = 1 )
=> C = 99 - 20 = 79 => C = 79
Similarly we will get the value of b
=> Solving F(t) for t = 15
=> F(15) = 20 + 79 x b15 = 60
=> 79 x b15 = 60 - 20 = 40 => b15 = 40/79 => b = (40/79)1/15
=> b = 0.95564265604
Now We need to calculate "t" so that F(t) = 40
=> 40 = 20 + 79 (0.95564265604)t
=> (0.95564265604)t = (20/79) = 0.2531645569620253
=> to find value of t we will apply log on both sides
=> Log (0.95564265604)t = Log (0.2531645569620253)
=> t x Log(0.95564265604) = Log (0.2531645569620253) { Logmn = n Log m}
=> t = Log (0.2531645569620253)/Log(0.95564265604) = (-0.59659709562/-0.01970447333) = 30.2772413973
=> It will take approximatley 30 Mintutes to Cool to 40 Degrees.
Happy To Help You
Waqar
Happy Chegging