A cup of hot chocolate has temperature 80 C in a room kept at 20 C.?
After a half hour the hot chocolate cools to 60 C.
a) what is the temperature of the hot chocolate after another half hour?
b) when will the chocolate have cooled to 40 C?
I think I know how to do b, I'm mostly struggling with part a. An explanation of how to do each steps would be much appreciated.
what is the function for determining temperature change?
linear, exponentially deacaying?
To answer both parts of the question, we can use Newton's Law of Cooling, which states that the rate at which an object cools is proportional to the temperature difference between the object and its surroundings.
Let's denote the initial temperature of the hot chocolate as Tc, the temperature of the room as Tr, and the rate at which it cools as k.
a) To find the temperature of the hot chocolate after another half hour, we can use the formula:
T(t) = Tr + (Tc - Tr) * e^(-kt)
where T(t) is the temperature at time t.
Given that the initial temperature T(0) is 80°C, the temperature of the room is Tr = 20°C, and the time elapsed is t = 0.5 hours, we can plug these values into the formula. However, we still need to find the value of k.
To find k, we can use the fact that the temperature of the hot chocolate at t = 0.5 hours is 60°C. Plug these values into the formula:
60 = 20 + (80 - 20) * e^(-0.5k)
Simplifying this equation will give us the value of k, which we can then use to find the temperature after another half hour.
b) To find when the chocolate will cool to 40°C, we set T(t) = 40°C and solve for t.
40 = 20 + (80 - 20) * e^(-kt)
Solving this equation will give us the time t at which the hot chocolate temperature reaches 40°C.
Hope this helps! Let me know if you have any more questions.