Calculus: Derive eqns for temp probe from Newton's Law, data points

DonaldMurray1

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I need help with the last page, I have been trying to do it for my spring break, but I just don't get it.



A temperature probe is placed in a cup of hot water. It remains there for approximately 35 seconds; then it is removed from this cup and placed in a cup of cool water for another 35 seconds. The objective of this activity is to find an algebraic function that models the temperature recorded by the probe over the entire 70 seconds, and then apply some calculus concepts to the function.

1. [A review of piecewise functions, not included here.]

Complete the chart below from the data that was collected from the experiment. The data appears to have horizontal asymptotes at y = 173 and y = 41.

Time0510152025303540455055606570
Temp86.9150.9165.6170.3172.0172.3172.5172.685.959.148.644.843.142.341.7

2. Create a scatterplot in your calculator. [Not included here.]

3. The General Solution (Newton's Law of Cooling/Heating)

We want to find a function F(t) that models the Fahrenheit temperature F of the probe at any time t, measured in seconds. Using a property of physics called Newton's Law of Cooling/Heating, the temperature in an activity such as this can be modeled by an exponential function in the form:

. . .F(t) = a bt + c

4. The Specific Solution

First, to find an algebraic function that models our temperature-versus-time data, it is clear that we need to write a piecewise function, one rule for the first (approximately) 35 seconds and another for the last (approximately) 35 seconds. To find both of these rules, we will use the property of Newton's Law of Cooling.

In order to find a model of the form F(t) = a bt + c for the first part of our data, we need to find the constants a, b, and c. We can find the constants a and c from the chart above, or by "tracing" on our scatterplot.

First, we need to find the constant c. According to Newton's Law of Cooling/Heating, and the data collected, the value of c would be approximately:

. . .c = ____________

To find a, record the temperature when t = 0: (0, _86.9_) Substitute this ordered pair, with the value of c, into our model F(t) = a bt + c, and solve for a.

. . .a = ____________

To find b, the last constant in the model, we need another ordered pair. Let's use the ordered pair from the chart when t = 10. Record the ordered pair: (10, ________)

Substitute these values into the equation (with the values of a and c), and solve for the last unknown constant b. Round b to three decimal places.

. . .b = ____________

. . .F(t) = ____________

To check your work, graph your equation with your scatterplot to see how it fits the first part of the data.

The equation that fits the second part of the data is similar to the first equation. However, since we are beginning with a time other than t = 0, we need to apply a "horizontal shift" to our function. Therefore, the resulting form of this function is:

. . .F(t) = a bt-h + c

where h = 35. Use this form, and the hints given for the first part of the function, and find a rule that fits this part of the scatterplot. Again, graph this equation to check it.

. . .a = ____________

. . .b = ____________

. . .c = ____________

. . .F(t) = ____________
 

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For us to help you, we need to know WHERE are you stuck specifically. Which part don't you understand, that is preventing you from filling in the blanks on the second page?

You have a model for the temperature vs. time (time is lowercase t).

\(\displaystyle \displaystyle \mathrm{temperature} = F(t) =a\cdot b^t + c \)

You can figure out which values of the constants a, b, and c apply to your specific situation (i.e. which values are a good model of your data) by substituting in the value for temperature that occurs at time t = 0:

\(\displaystyle \displaystyle 86.9^\circ\mathrm{F} = F(0) = a\cdot b^0 + c \)
 
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Okay, re-reading it, it's actually a little bit tricky, in the sense that they give you a non-intuitive form for the function.

Here are some additional hints that might help:

- plot a simple exponential function like \(\displaystyle e^t\) using your graphing calculator. You'll see something that is always rising, slowly at first, but then ever more quickly. It just blows up and has no upper limit. This is not at all like what your data for the heating phase of the experiment look like.

- Now try plotting something more like \(\displaystyle 1 - e^{-t} \). In this case, the function rises, quickly at first, but then more slowly, until it reaches some asymptotic value (the 1, in this case). This looks a lot more like your data during the heating phase.

1) What does this tell you about the nature of c? How can you infer it from your data? Hint: what would be the asymptotic value if you replaced the 1 with a c in the equation above?

2) What does this tell you about the sign of a? Hint: the function that looks kind of like your data is \(\displaystyle 1 - e^{-t} \), with a minus sign, not a plus sign.

3) What does this tell you about the nature of b? Hint: the function that looks kind of like your data has \(\displaystyle e^{-t} = 1/e^t \). So, in order to match this form, does b have to be > 1, or < 1?
 
A temperature probe is placed in a cup of hot water. It remains there for approximately 35 seconds; then it is removed from this cup and placed in a cup of cool water for another 35 seconds. The objective of this activity is to find an algebraic function that models the temperature recorded by the probe over the entire 70 seconds, and then apply some calculus concepts to the function.

1. [A review of piecewise functions, not included here.]

Complete the chart below from the data that was collected from the experiment. The data appears to have horizontal asymptotes at y = 173 and y = 41.

Time0510152025303540455055606570
Temp86.9150.9165.6170.3172.0172.3172.5172.685.959.148.644.843.142.341.7

2. Create a scatterplot in your calculator. [Not included here.]
Would it be correct to gather that you and fellow students ran this experiment, and that the data are derived from that experiment?

4. The Specific Solution

First, to find an algebraic function that models our temperature-versus-time data, it is clear that we need to write a piecewise function, one rule for the first (approximately) 35 seconds and another for the last (approximately) 35 seconds. To find both of these rules, we will use the property of Newton's Law of Cooling.

In order to find a model of the form F(t) = a bt + c for the first part of our data, we need to find the constants a, b, and c. We can find the constants a and c from the chart above, or by "tracing" on our scatterplot.

First, we need to find the constant c. According to Newton's Law of Cooling/Heating, and the data collected, the value of c would be approximately:

. . .c = ____________
My guess would be the temperature(s) when the probe was about to be put into one or the other cup, as this would likely be "room temperature".

To find a, record the temperature when t = 0: (0, _86.9_) Substitute this ordered pair, with the value of c, into our model F(t) = a bt + c, and solve for a.

. . .a = ____________
Plug-n-chug, and see what you get.

To find b, the last constant in the model, we need another ordered pair. Let's use the ordered pair from the chart when t = 10. Record the ordered pair: (10, ________)
Copy this from the table.

Substitute these values into the equation (with the values of a and c), and solve for the last unknown constant b.
You will have an equation of the form:

. . . . .F(10) = [value of a] b10 + [value of c]

Plug in the known values of F(10), "a", and "c". Solve for "b".

If you get stuck, please reply showing your thoughts and efforts. Thank you! ;)
 
Would it be correct to gather that you and fellow students ran this experiment, and that the data are derived from that experiment?


My guess would be the temperature(s) when the probe was about to be put into one or the other cup, as this would likely be "room temperature".

Hi stapel, FWIW, I don't think that c would be the initial temperature prior to putting the probe in a cup. I say that because plugging t = 0 into the equation doesn't give you c. As I was trying to hint to the OP in my response, I think c is the steady-state temperature.
 
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