Newton’s Law of Cooling A cup of coffee is removed from a micr...

Question

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.
c. When does the temperature of the coffee reach 50°C?

c. When does the temperature of the coffee reach 50°C?

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us highlight all the key pieces of information that we need to use in order to solve this problem. A glass of hot tea is taken out of the kettle at a temperature of 90 °C and left to cool in a room where the temperature is 20 °C. After 10 minutes. The tea has cooled to 70 °C. At what time, after being taken out of the kettle, will the tea cool to 50 °C? Round your answer to the nearest minute. Awesome. So it appears our final answer that we're ultimately trying to solve for is how long in units and minutes, noting that we're rounding to the nearest whole minute. How long will it take the tea to cool to 50 °C, noting that that this is after it's taken out of the kettle. So how long will it take the tea to cool to 50 °C? So how long in units and minutes? So now we know what we're ultimately trying to solve for. Our first step that we need to take in order to solve this problem is we need to recall and use Newton's law of cooling, and Newton's law of cooling states that capital T of lowercase t is equal to TA plus parentheses T0 minus TA. All multiplied by E to the power of negative K multiplied by T. Where TA denotes the ambient temperature, T0 denotes the initial temperature. And K is greater than 0 represents the cooling constant. Awesome. And let us know that this capital T denotes the temperature. Fantastic. So for our problem, let us note that TA, the ambient temperature, is equal to 20 °C. So now that we know that the ambient temperature is 20 °C, and this is given to us by the problem itself. We're also told that the initial temperature T0 is equal to 90 °C, and T of 10. Where we're told that the minutes after 10 minutes, so that's where T, this lowercase t denotes the time. So this is our function for the temperature. So capital T of lowercase t where T once again is our temperature, lowercase t is the temperature. So when we plug in a value of 10 in for T, T at 10 will equal 70 °C. Awesome. So now our second step is we need to write the temperature function. So capital T of lowercase t is equal to 20 + 90 minus 20 multiplied by E to the power of negative K multiplied by T, which will be equal to 20 + 70 multiplied by E to the power of negative K multiplied by T. Fantastic. Our third step is we need to recall and use the data. T T of 10 is equal to 70 to help us find the value for K. So when we plug in our value. Of 70, we will find that 70 is equal to 20 plus, so it's gonna be 20. Plus 70 multiplied by e to the power of -10 multiplied by K, which we, when we start to rearrange, we will find that it'll be, we can write that 50 is equal to as we move the 20/50 is equal to 70 multiplied by E to the power of -10 multiplied by K. So let us know that we are ultimately trying to solve for K all by itself, so we need to rearrange this expression to isolate and solve for K. So in an effort to save time, I'm going to skip a few steps, but you're going to use basic algebra to isolate and solve for K. And when you do, you'll find that K is simply equal to 1 divided by 10, multiplied by the natural log of 7. It's gonna be equal to the natural log of 7 divided by 5. Fantastic. So with that in mind, our next step, since we solve for K, our next step is we need to find the time T, specifically lowercase t, because lowercase t once again represents the time. So we need to find the time lower case t when capital T of lowercase t is equal to 50 °C. So let's make a note of that where T of T is equal to 50 °C. So, with that in mind, if we plug in a value of 50, we will find that 50 is equal to 20 plus 70, multiplied by E to the power negative K multiplied by T. And when we move the 20 over to the other side, it means we subtract 20 from one side to move it to the other side, the left side of the equation, we will get 30 is equal to 70 multiplied by E to the power negative K multiplied by T. Fantastic. So now, We are told That we're trying to solve for T. So once again, let's skip a few steps pieces, you're going to just use basic algebra again to rearrange this expression to isolate and solve for T. And when you do, you will find that lowercase t at the time, is going to be simply equal to drum roll, the natural log of 7 divided by 3, divided by K. Fantastic, and that is what our value of lowercase t will be equal to. Fantastic. So now, our next step. we need to note that, so once again, we know our value for T, but we need to solve for the numerical value. So we found the expression for T, I should be specific. We found this expression, this particular expression for T, but we need to find in units of minutes how long time wise we're dealing with, or how long it takes time wise for this T to cool down. So with that in mind, Our next step now is we need to substitute this value of K is equal to 1 divided by 10, multiplied by the natural log of 7 divided by 5 into our expression, and when we do, we will find that T is equal to the natural log of 7 divided by 3, divided by 1 divided by 10 multiplied by the natural log of 7 divided by 5. Which will be equal to 10 multiplied by the natural log of 7 divided by 3 divided by the natural log of 7 divided by 5, which is approximately when we round to the nearest full number, it will be approximately equal to 25, not forgetting that our units are in minutes, and that's it. That is our final answer. Hooray, we did it. So looking back at the prom itself to recap what we've learned, our final answer now, without a doubt, we discovered will have to be 25 minutes. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.
c. When does the temperature of the coffee reach 50°C?

Key Concepts

Newton’s Law of Cooling

Exponential Decay Model

Solving for Time in Cooling Problems

Watch next

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C. c. When does the temperature of the coffee reach 50°C?

Key Concepts

Newton’s Law of Cooling

Exponential Decay Model

Solving for Time in Cooling Problems

Watch next

Newton’s Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80°C and allowed to cool in a room with a temperature of 25°C. Five minutes later, the temperature of the coffee is 60°C.
c. When does the temperature of the coffee reach 50°C?