A student observes that the temperature of some water placed on a hot plate increases over a two-minute time interval. The student concludes that the temperature increase of the water will be 50 times greater if the water is left on the hot plate for a time interval that is 50 times longer. Which of the following statements about the student’s conclusion is correct?
Okay, before we get to these statements, let’s consider what’s going on here. We have some water. Say it’s in a container like this. And this water has been placed on a hot plate, a surface that can be programmed to get very hot, over a time interval of two minutes. The student observes that the temperature of the water goes up. And based on this observation, the student draws a conclusion. The conclusion is that the temperature increase of the water would be 50 times greater if the water was left on the hot plate 50 times longer. So let’s say that over this two-minute time interval, the water temperature went up two degrees Celsius. The student’s conclusion is that if the water was left on 50 times longer, that is for 100 minutes, then its temperature increase would be 50 times this two-degree Celsius amount. It would be 100 degrees. So that’s the conclusion that the student drew.
And now, let’s consider some statements about this conclusion. Option A) The student’s conclusion is correct. Option B) How much the temperature of the water changes depends on how much the water temperature is higher or lower than the hot plate. The more the water increases in temperature, the slower it will continue to increase in temperature. And option C) The water will not increase in temperature after two minutes.
So of these three options, of these three explanations of the student’s conclusion, we want to pick which one is correct. One way we can investigate these conclusions is by putting specific numbers to the temperature of the hot plate and the temperature of the water when it starts out. Of course, we don’t know that these temperatures are accurate. But just having particular numbers to work with can help clarify our thinking. Let’s say that at the outset, the temperature of the water was 20 degrees Celsius. And we can say further that the hot plate was set to a temperature of 65 degrees Celsius. So naturally, if we put 20-degree water on a 65-degree hot plate, the temperature of the water will go up. But we can also see that it won’t go up forever. That’s because the water temperature will never get higher than the temperature of the hot plate that’s heating it up. Once these two temperatures are the same, we wouldn’t expect the water to get any hotter.
In the experiment, the student waited two minutes and observed that the water temperature went up. Let’s say that at the end of those two minutes, the temperature of the water was 25 degrees Celsius. And again, this is just a made-up number. But it still could be helpful to us. Taking these temperature values that we’ve come up with, let’s create a plot showing the temperature of two things: the hot plate and the water over time.
To do this, we can create two axes. The horizontal one shows time in minutes, and the vertical one indicates temperature in degrees Celsius. When we put in values on these horizontal and vertical axes, we can do it this way. The temperature on our vertical axis goes up to 65 degrees Celsius, and then it goes down to 45, 25, five. And then, there’s a break. And we get to our origin. At the origin, both the temperature and the time are zero. Then, considering our time axis, again, that axis starts at zero minutes, but then goes up to one minute, two minutes. And then there’s this break and we jump up to 100 minutes. The reason for these breaks on the horizontal and vertical axes is because we just want to plot a certain range of values, the values relevant to our experiment.
And now, we’re ready to start plotting in the temperatures of our hot plate and water. Let’s start with the temperature of the hot plate. We’ve said that at time 𝑡 equals zero, that temperature is set to 65 degrees Celsius. And we say that it stays constant all throughout the experiment. So there’s our hot plate temperature holding steady at 65 degrees. Then, the water temperature we know starts out at 20 degrees Celsius. That’s at time 𝑡 equals zero minutes. And we’ll say that that’s at there. And then that over the course of two minutes, that temperature increases to 25 degrees Celsius. Plotting that point on our graph, it would be at two minutes and 25 degrees Celsius.
So we can see that the water temperature has gone up over this interval. And here’s something interesting. If we were to draw a line between these two data points for the temperature of the water at zero minutes and two minutes, it wouldn’t be a straight line directly from one point to the other. The reason is the rate at which water temperature increases is not constant. The farther away the temperature of the water is from the temperature of the hot plate, the faster the water temperature will increase.
So this means that rather than being a straight line like this, the temperature of the water will instead go up faster and then continue to increase but not at the same rate. Its rate of increase will slow as its temperature rises. And this has to do with the fact that the water temperature, like we noted before, will never exceed the temperature of the hot plate. So as it approaches that temperature, it approaches it more and more slowly.
Knowing all this, let’s start by evaluating answer option C, which claims that the water will not increase in temperature after two minutes. Well, just looking at the situation, as well as seeing the gap in temperature between the hot plate and the water that remains after two minutes, we can say that conclusion C is not correct. In general, even if we didn’t know the particular temperatures of the water and the hot plate over time, we couldn’t rule out the possibility that the water temperature was not yet as hot as the hot plate. In other words, that it would continue to increase. So we can’t say that option C is correct.
What about option A? This option says that the student’s conclusion, which is that if the water was left on the hot plate 50 times longer, then its temperature increase would be 50 times greater is correct. For option A to be correct, two things would have to be true. Number one, the water temperature would need to increase at a steady, constant rate. But we’ve seen that that’s not the case. The water temperature increases faster the farther away its temperature is from the temperature of the hot plate. But as the temperatures get close to one another, that rate decreases. But then, another thing that would have to be true is that over this 100-minute time interval, 50 times two minutes, the temperature of the water couldn’t have reached the temperature of the hot plate. That’s because if it did, there would be nothing to make it get any hotter. Its temperature would not continue to increase.
Of these two conditions, we know the first one won’t be met because the temperature increase rate isn’t constant. And we can’t be certain in general about the second one. For all we know, the water temperature would have heated up to the hot plate temperature long before 100 minutes had elapsed. So we can’t say that option A is a correct interpretation, either.
What about option B? This says how much the temperature of the water changes depends on how much the water temperature is higher or lower than the hot plate. It goes on to say, the more the water increases in temperature, the slower it will continue to increase in temperature. This answer basically breaks up into two separate statements: statement one is this part here and statement two is this part here. And what we’ve seen so far supports both statements. We’ve seen the rate of water increase isn’t constant. And we’ve also seen that the total amount of water temperature change depends on how different that initial temperature is from the temperature of the hot plate. This, then, is the correct interpretation of this experiment.
How much the temperature of the water changes depends on how much the water temperature is higher or lower than the hot plate. And the more the water increases in temperature, the slower it will continue to increase in temperature.