As part of his summer job at a restaurant, George learned to cook a large pot of
ID: 3146763 • Letter: A
Question
As part of his summer job at a restaurant, George learned to cook a large pot of soup late at night, just before closing time, so that there would be plenty of soup to feed customers the next day. He also found out that, while refrigeration was essential to preserve the soup overnight, the soup was too hot to be put directly into the fridge when it was ready. (The soup had just boiled at 100 degrees Celsius, and the fridge was not powerful enough to accomodate a big pot of soup if it was any warmer than 20 degrees Celsius). George discovered that by cooling the pot in a sink full of cold water, (kept running, so that its temperature was roughly constant at 5 degrees Celsius) and stirring occasionally, he could bring that temperature of the soup to 60 degrees Celsius in ten minutes. How long before closing time should the soup be ready so that George could put it in the fridge and leave on time?Explanation / Answer
Here let say the temperature of soup at perticular time t is T. Here we can say that the change in temperature of soup with time is proportional with the temeperature difference of soup and water. Mathemetically,
dT/dt = -k(T -T0) where T0 = 5
dT/ (T -5) = -kdt
ln (T -5) = -kt + C1
T - 5 = C0 e-kt
T = 5 + C0 e-kt
at t = 0, T = 100
100 = 5 + C0
C0 = 95
and at t = 10 mins , T = 600 C
60 =5 + 95 * e-k*10
55 = 95 * e-10k
e-10k = 0.5789
10k = 0.5465
k = 0.05465
so,
T = 5 + 95 e-0.05465t
so let say we will wait for t = t0 sec , T would be 20o C
20 = 5 + 95 * e-0.05465 * t
15/95 = e-0.05465 * t
0.1579 = e-0.05465 * t
-1.8458 = -0.05465 * t
t = 33.775 minutes
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