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KA
Numerade Educator

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Problem 14 Easy Difficulty

Suppose you have just poured a cup of freshly brewed coffee with temperature $ 95^{\circ} $ in a room where the temperature is $ 20^{\circ}. $

(a) When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain.

(b) Newtons Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Write a differential equation that expresses Newtons Law of Cooling for this particular situation. What is the initial condition? In view of your answer to part (a), do you think this differential equation is an appropriate model for cooling?

(c) Make a rough sketch of the graph of the solution of the initial-value problem in part (b).

Answer

(a) At the beginning.
Slows down.
(b) $-\frac{d T}{d t}=k(T-20)$
(c) See Graph

Discussion

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Video Transcript

for part A. The coffee cools down most quickly in the beginning, when it is taken off the heat source, and as time goes on, the rate of cooling will slow down as it approaches. Room temperature are what we will be referring it thio as the ambient temperature heartbeat. Newton's all of cooling states at the rate of cooling oven object is proportional to the difference between the temperature oven object and the ambient temperature To write that in mathematical terms going to say the rate of change in temperature in respect of time should be proportional to the temperature oven object in the ambient temperature. Gonna put a negative in front there to say that it is cooling. Do you not like that negative With the K, we can always rewrite it like this. The only thing we have left to do now is plug in the ambient temperature. We're also asked what the initial temperature is. Just 95 degrees Celsius. We can express this just like this. Part C. We're gonna do a rough sketch of a graph 20 95 going to draw a dotted line just to make sure I don't go past my ambient temperature. We're going to say yes. The model in the answers support each other because as the temperature approaches the ambient temperature of 20 the rate of change in temperature with respect to time is approaching zero. Thank you so much.