44. Silver cooling in air The temperature of an ingot of silve...

Question

44. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be
b. 2 hours from now?

b. 2 hours from now?

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A loaf of bread is 90 °C above room temperature now. 15 minutes ago it was 120 °C above room temperature. According to Newton's law of cooling, how far above room temperature will the bread be 45 minutes from now? Round your answer to the nearest degree. OK. So it appears for this particular prompt we're asked to take all the information that is provided to us and considering Newton's law of cooling we're asked to determine how far above room temperature will this loaf of bread be 45 minutes from now. And we're asked to take our final answer, and we're asked to round to the nearest whole degree. So that means our final answer isn't going to have a decimal in the answer, so we're not going to use a decimal, it has to be a whole number. So, we're rounding it to the nearest degree. So with that said, now that we know what we're ultimately trying to solve for, in order to solve for this problem, we can recall a note that we can use a simplified version of Newton's law of cooling. And since the problem asks for the temperature above room temperature, we can define DT as the temperature difference. So DT will be equal to D0 multiplied by E to the power of K multiplied by T. Where D of T is the temperature difference at time T. And D0 is the initial temperature difference, and K is our cooling constant. So our first step now is we need to identify our values that are given to us for the prom itself. We need to set the now time T as T is equal to 0, and let's make a note that this is for the now time. And we need to note that now T is equal to 0, which will mean that D of 0 will be equal to 90 °C. And 15 minutes ago, that will mean when T is equal to 15, that will mean that D of 15 will be equal to, so once again, this will be equal to 120 °C. And once again, we're ultimately trying to determine how far above room temperature will this loaf of bread be in 45 minutes from now. So that will be our goal, which is T is equal to 45, meaning we need to solve for D of 45, and D of 45 is unknown, and that's our unknown value that we're ultimately trying to solve for. What is D of 45? So with that in mind, our second step now is we need to determine the decay ratio. So instead of solving for our constant K directly, once again, this is our cooling constant is K, and instead of solving for K directly, It is often more easier to find the ratio of cooling over a specific time interval. So let's look at the change over 15 minutes, meaning we need to take D 0 divided by D of 15, which will be equal to 90 divided by 120, which will be equal to 3 divided by 4 or 3/4. Fantastic. And that means that for every 15 minutes, the temperature difference becomes 75% of what it was at the start of that specific interval. So with that in mind, moving right along here, our third step. In our final step, so this is our 3rd and final step. We now need to calculate what the temperature of this loaf of bread will be 45 minutes from now. So the time interval from T equals 0 to T equals 45 is exactly 3 15 minute periods, meaning we need to take 45 divided by 15, which is equal to 3. So we can therefore, based on our prior knowledge, we can recall and apply the decay ratio 3 times. So D of 45 is equal to D 0 multiplied by 3 divided by 4 to the power of 3. So plugging in our known variables, we will find that D of 45 will be equal to 90. And we can leave off our units for now, so we could just leave them as numbers for a moment. So we'll take 90 multiplied by parentheses of 27 divided by 64, because 3 to the power of 3 is 27 and 4 to 3 is 64. And when we plug this expression into our calculator and we round to the nearest whole degree, we will find that it's approximately equal to 38 °C. And when you type it into your calculator initially, you'll find that it's equal to 37.96 °C, which will mean when you round to the nearest whole degree, it'll be approximately equal to 38 degrees. Celsius and that is our final answer. hooray, we did it. So looking at the prom itself to recap what we've learned, we could safely say that our final answer for the how far above room temperature this loaf of bread will be 45 minutes from now will be. Approximately 38 °C, and that's it, that's our final answer. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video. Bye.

44. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver beb. 2 hours from now?

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44. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be
b. 2 hours from now?