A community for students.
Here's the question you clicked on:
 0 viewing
anonymous
 4 years ago
1. Newton’s law of cooling states that the rate of cooling of an object is proportional to
the difference between its temperature and the ambient temperature.
(a) Assuming this, formulate an initial value problem that models the cooling of a
cup of coffee.
(b) Suppose that the ambient temperature is 20◦ C, the coffee is initially at a temperature
of 90◦ C, and is initially cooling at a rate of 8◦/minute. How long will
it take for the coffee to cool to 75◦.
anonymous
 4 years ago
1. Newton’s law of cooling states that the rate of cooling of an object is proportional to the difference between its temperature and the ambient temperature. (a) Assuming this, formulate an initial value problem that models the cooling of a cup of coffee. (b) Suppose that the ambient temperature is 20◦ C, the coffee is initially at a temperature of 90◦ C, and is initially cooling at a rate of 8◦/minute. How long will it take for the coffee to cool to 75◦.

This Question is Closed

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0i believe the answer is1.875 minutes

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0can someone instruct me how to get the answer ...

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0Let \( T_A \) be the ambient temperature and T the temperature of the object cooling. What the problem tells you is \[ \frac{dT}{dt} \ \ \ \alpha \ \  (T  T_A) \] hence you can write down the ODE with a constant of proportionality.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0Now. Use the facts given to you in part (b) to calculate that constant. Then, solve the ODE and apply the initial values given to obtain the solution to this problem. Finally, find the time t such that T = 75 C.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0So, does what I wrote down above make sense and do you see what to do next?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0not really .. i dont think that we're supposed to use the second part of the question to figure out part a.. and i'm not sure where the alpha came from

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0alpha means "is proportional to"

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0The change in temperature of the object, like a cup of hot water, is proportional to the difference of its temperature to the ambient temperature. The negative sign is there because the temperature is going down towards the ambient temperature if it currently above it; or changing upwards, if T < T_a.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0hmmm.. i wish i understood, thanks anyways though

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0Let's try this one more time. You're told that: "Newton’s law of cooling states that the rate of cooling of an object is proportional to the difference between its temperature and the ambient temperature." Let's write T(t) for the temperature of the cooling object. This law of cooling is telling us something about \[ \frac{dT}{dt} \] so far so good?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0ok i think i may have misread the question ... it's asking for an inital problem and not initial formula .. i'm not really sure what it's asking.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0Now, what is the change proportional to? "proportional to the difference between its temperature and the ambient temperature" Let \( T_A \) be the ambient temperature, then the difference between the temperature T and the ambient temperature is \[ T  T_A.\] Hence \[ \frac{dT}{dt} \ \text{ is proportional to } (TT_A) \] (We write \[ \frac{dT}{dt} \ \ \alpha \ (TT_A) \] this is standard notation; I'm sure you've seen it before, but it isn't very important for this problem.)

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0What is important is the negative sign. It must be negative because if \( T > T_A \) then T should be going down. I.e., the object is cooling and its temperature T should be getting closer to the ambient temperature. So far, so good? Be honest.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Ok if I'm being honest, I just started a differential calculus class and I don't even know what we're learning .. I'm not even sure what the objective of the course is. I sound stupid, but promise just takes a little time ahah .. Ok, so for the first part all we're trying to say is that the 2 are proportional to eachother? we dont need an equal sign or anything then in the answer right?

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0We do. We now introduce a constant of proportionality. What that equation means in concrete terms is \[ \frac{dT}{dt} = k(TT_A) \] for some positive constant k. This is the ODE we want.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0ODE means original differential equation?

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0ordinary differential equation.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0lmao alright. you must think i'm a retard.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0I'm going to refer you now to this lecture where this equation is solved. http://ocw.mit.edu/courses/mathematics/1803differentialequationsspring2010/videolectures/lecture3solvingfirstorderlinearodes/ I *strongly* recommend you watch it, take notes, rewind, watch it again. We all take a while to understand new concepts. This example of Newton's Law of Cooling is a classic.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0after you've watched it, if you still have doubts, come find me.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0ok .. will do, thanks

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0btw, the solution method he shows you in the lecture is more powerful than what you need. But no harm done, because it is a good general method. I can still show you a simpler method for you equation in particular: separation of variables. You may know it already.

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0but more than "it's a good general method", this is going to be the next thing you learn in your class anyway.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0kind of... i'm working on b now, and for solving ... are you able to walk me through the steps to getting the solution?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0if your still there ... i got T=Ta +_ e^(kt+c)

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0Right, \[ T(t) = T_a + Ae^{kt} \] For some constant A ( = e^c ) Now, in part b, we can solve for A. We have that T(0) = 90. Therefore \[ 90 = T(0) = T_a + A \] hence, \[ A = 90  T_a \] We also know that \( T_a = 20 \), thus \( A = 90  20 = 70 \) and the solution is \[ T(t) = 20 + 70e^{kt} \] ok?

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0yes thank you so much!

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0Now, one more thing. You need to find k. How are yo going to do that? You're given one more piece of information in (b) "[the coffee] is initially cooling at a rate of 8◦/minute". Translate that into the mathematics.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0so in math terms thats .. 8t?

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0"Cooling at a rate of 8 degrees/minute" That must mean dT/dt at t = 0 is 8 degrees/minute Now dT/dt = k(T_a  T) You're told that for t = 0 dT/dt = 8 degrees/minute We also have that k(T_a  T) = k(2090) = 70k Thus 70k = 8 You can now solve for k.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0ohhh .. ok, and then once i've solved for k i can solve for t?

JamesJ
 4 years ago
Best ResponseYou've already chosen the best response.0Yes, then finally you can find the t for which T(t) = 75

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0aren't you just wonderful, thank you so so so much!
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.