SOLUTION: A pan of water is brought to a boil and then removed from the heat. Every 5 minutes thereafter the difference between the termperature of the water and room temperature is reduced

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Question 1188251: A pan of water is brought to a boil and then removed from the heat. Every 5 minutes thereafter the difference between the termperature of the water and room temperature is reduced by 50%.
a) Room temperature is 20 degrees Celcius. Express the temperature of the water as a function of the time since it was removed from the heat.
b) How many minutes does it take for the termperature of the water to reach 30 degree Celcius.
Found 2 solutions by Solver92311, ikleyn:
Answer by Solver92311(821) About Me  (Show Source):
You can put this solution on YOUR website!



Please post your Physics problems on a Physics website

John

My calculator said it, I believe it, that settles it

From
I > Ø

Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
A pan of water is brought to a boil and then removed from the heat.
Every 5 minutes thereafter the difference between the temperature
of the water and room temperature is reduced by 50%.
a) Room temperature is 20 degrees Celsius. Express the temperature of the water
as a function of the time since it was removed from the heat.
b) How many minutes does it take for the temperature of the water to reach 30 degree Celsius.
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            It is a  STANDARD  problem on the  Newton's law of cooling,
            which is a standard application of logarithms and exponential functions.

            So,  it is  TOTALLY  GOOD  ( legal )  for this forum. 


I will assume that the boiling temperature is 100 °C (standard for water at normal conditions).


Then according to the Newton's cooling law, the temperature of the water at any time moment t 
after removing from the heat is


    T(t) = 20 + %28100-20%29%2A%281%2F2%29%5E%28t%2F5%29.


This formula is the answer to your first question (a).



To answer your second question, write this equation as you read the problem


    30 = 20+%2B+80%2A%281%2F2%29%5E%28t%2F5%29.


Simplify it step by step


    30 - 10 = 80%2A%281%2F2%29%5E%28t%2F5%29

      20    = 80%2A%281%2F2%29%5E%28t%2F5%29

      20%2F80 = %281%2F2%29%5E%28t%2F5%29

      1%2F4 = %281%2F2%29%5E%28t%2F5%29.


At this point, the answer is OBVIOUS:  t%2F5 = 2,  or  t = 2*5 = 10 minutes.


It is the answer to your second question (b).

Solved and thoroughly explained in all details.

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To see many other similar  (and different)  cooling problems,  look into the lesson
    - Solving problem on Newton Law of cooling
in this site.