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### Course: Differential equations>Unit 1

Lesson 5: Exponential models

# Worked example: Newton's law of cooling

The general function for Newton's law of cooling is T=Ce⁻ᵏᵗ+Tₐ. In this video, we solve a word problem that involves the cooling of a freshly baked cookie!

## Want to join the conversation?

• This may be a dumb question, but why isn't T(0), not t(0), if we are talking with respect to time?
• Not dumb at all! Never fear asking a question. T(t) is our function, Temperature with respect to time, and so when asking what T(0) is, we are asking what the Temperature is at time 0. Just like if we have a function f(x) and we plug in x=5, we will have f(5) and not x(5). Hope that helped!
• Does Newton's Law of Cooling only work in degrees Celsius? could we use Fahrenheit or even Kelvin?
• You can actually use any measure of temperature with newtons law of cooling because it deals with temperature generally (no units). Its the same for the time variable. In his example, Sal uses an arbitrary 2 to represent 2 mins. That could actually represent 2 days, weeks, hours, or years. Essentially, then, what you get out of the equation for units is what you put in it.
To test this for yourself, try doing the problem over again but convert all of Sal's measurements to Fahrenheit and see if the answer works out to the same amount of cool down time (Hint: it does).
• Early on in the video, Sal states the assumption that the ambient temperature will not change. How would solving this change if the ambient temperature was not constant?
``dT/dt = k(T-A(t))``

where A is a function of time corresponding to ambient temperature. For example, if temperature increases linearly, A = mt, where m is a constant.
• I have a question rather than putting the negative in front of the "k" could you just switch the (T-Ta) to (Ta-T)?
• Yes, that is also valid. But historically the equation has been solved with a negative `k`, so that's why it's taught that way.
• when do you know when to take the absolute of a natural log and when not to?
• When integrating 1/x, you always get the natural log of the absolute value of x. If x is going to always be positive or always negative, then you can remove the absolute value and replace it with just x or just -x.
• If we use the Law of Cooling to describe the temperature at any moment, then when will the temperature of the oatmeal be the same as that of the environment?
• If you calculate t for T(t)=20.01, which is very close to the ambient temperature, you'll find 42.9 minutes. If you set T(t)=20, you'll notice it indeed can never happen as there's no t that can make exp(t*ln(2/3)/2)=0.
• How do you use this to find what temperature something will be at certain time instead of the time it will become a certain temperature?
• With known initial and ambient temperatures, you can use the T1 = A + Te^rt in two ways: if you know the rate of change AND the time, you can just plug both r and t into the equation to get T1 (the temperature you're looking for). What Sal did was just solve in the other direction; he used a known T1 to find the corresponding t.

Take this example: 50+30e^(-.5t)=T1. As r is already known to be -.5, you can plug in any value of t that you want and get a temperature.