27–30. Newton’s Law of Cooling Solve the differential equation...

Question

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.

A pot of boiling soup (100°C) is put in a cellar with a temperature of 10°C. After 30 minutes, the soup has cooled to 80°C. When will the temperature of the soup reach 30°C

Accepted Answer

Welcome back everyone. In this problem, a metal rod at 150 °C is left in a laboratory at 25 °C. After 20 minutes, its temperature drops to 100 °C. How long will it take for the rat to reach 50 °C? On their answer to the nearest minute. A says 42 minutes. B 63 minutes. C 84 minutes, and D 93 minutes. Now, how can we figure out how long it will take for the rod to reach 50 °C? Well, we're talking about temperature here and recall that by Newton's law it tells us that the temperature as a function of time is equal to the environment's temperature, in this case the temperature of the laboratory, plus. The original temperature T knot minus the environment's temperature, OK, multiplied by E to the power of negative KT where K is a constant. Now in this case if we think about it. We know that initially we know that our environmental temperature, the temperature of the laboratory is 25 °C. Initial temperature T0 is 150 °C, and we know that after 20 minutes the temperature drops to 100 °C. So the temperature as a function of time T equals 20 is equal to 100 °C. So how can we use all of this to figure out how long it takes for the rod to reach 50 °C? Well, let's use T and TN to get an expression or an equation for our temperature as a function of time. See if we can use what we know about our temperature dropping to 100 °C after 20 minutes to figure out the value of our constant in that equation and then use that to figure out the time it takes for the rad to reach 50 °C. So I mentioned 3 things there. Let's take it step by step. First, let's use what we know in our format. Now applying what we know, this means, OK, that. From T of 20 equals 100 °C, then 100 is going to be equal to 25 plus 150 minus 25 multiplied by E to the pore of negative 20k because T would be 20 and the T of T would be 100. T20 is 100. Now, if we try to simplify here, first we can subtract 25 from both sides, so that leaves us with 75 on the left hand side and on the right, 150 minus 25 is 125. So 75 equals 125 E to the negative 20K. If we divide both sides by 125, then we should get E to the negative 20K to be equal to 35, OK? Because 75 divided by 125 simplified is 3/5. And now if we use what we know about logarithms, then if either the negative 20K equals 35, that means negative 20K is equal to the natural log of 3/5. Therefore, Make some space here. This tells us that K will be equal to. 1/20, OK, multiplied by LN of 3/5. Now this is pretty good because we have a value fork that we'll be able to use in our equation. Because no, this tells us then that for tea of tea. Ealing 50 °C, OK. This implies that 50 will be equal to 25. OK. Plus. 125 E to the negative K. We know our value of K. I'll, I'll write it in later, but for now, let's just leave it as K. But it is of E to the negative K T. Now if we subtract 25 from both sides, then 25 equals 125 e to the negative kT, which means that e to the negative kT equals 15th when we divide through by 125. Again, if we apply what we know about logarithms, the definition of logarithms, that means negative K T equals the natural log of 15th. And remember, no, at this point we can put our value of K or substitute our value of K into our problem. So that means the negative value of -120 LN 3/5. OK. Multiplied by T will be equal to a length of 1/5. This means then negative multiplied by negative equals are positive, so T will be equal to LN 15th divided by 1/20 of 1/20 multiplied by LN of 3/5. And thus We could rewrite this then to say T equals 20 multiplied by LN of 15th, which is approximately 1.6094 divided by LN of 35, which is approximately negative 0.5108. And no, this meanst must be approximately equal to 63 minutes. Get to the nearest minute. Thus the rod will reach 50 °C after approximately 63 minutes. Thanks a lot for watching everyone. I hope this video helped. That means sorry, just to recap, B would be the correct answer.

Key Concepts

Newton’s Law of Cooling

Solving First-Order Differential Equations

Applying Initial Conditions and Solving for Time

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