Boiling Water
By
Janet Shiver
EMAT 6680
Investigation: Take a cup of boiling water and measure its initial temperature. Then record the temperature of the water each minute for thirty minutes. Graph the data and then construct a function to model the data. Calculate a measure of the error between the model and the observed data. Finally, using the function, predict the temperature after 45 minutes, 60 minutes, or 300 minutes.
Collect the Data
After boiling a cup of water, I recorded its temperature every minute for thirty minutes. The data is graphed below, with the independent variable as time in minutes and the dependent variable as temperature.
Time 
Temperature 


0 
210 
1 
200 
2 
189 
3 
180 
4 
174 
5 
167 
6 
161 
7 
156 
8 
151 
9 
148 
10 
145 
11 
142 
12 
139 
13 
136 
14 
134 
15 
131 
16 
128 
17 
126 
18 
125 
19 
123 
20 
121 
21 
119 
22 
117 
23 
116 
24 
114 
25 
113 
26 
112 
27 
110 
28 
109 
29 
108 
30 
107 
Finding a model
Our next task will be to develop a model for this data. We must attempt to find a cooling function that will “best” fit our data. As we do this, we must keep in mind that since our data is “realworld” it is highly unlikely that we will be able to find a curve that matches it exactly. We need to focus instead on finding a function that will best match our data.
We will begin by first examining the general shape of the graph and its behavior. Several basic functions can be eliminated based on the appearance of the graph. The graph of the data does not lie on line so the linear function can be eliminated as a possibility. It is also does not resemble a parabola so the quadratic function would not be a good choice. We can also eliminate the cubic function and higher order polynomials for the same reasons of appearance. So we can safely say that the graph does not resemble any polynomial functions. It is also safe to eliminate the trigonometric functions since the graph has no periodicity.
The function that the graph most closely resembles is the exponentional decay function. As with all exponential decay models, our graph shows a rapid decrease at the beginning of the time. As time progresses, the graph continues to decrease in temperature but at a much slower rate. We could conjecture that if we continued to take readings that the water would ultimately reach room temperature which in this case was 77 degrees.
Exponential Decay Model
The general form of any exponential model is , where a is the initial value at time 0, k is the decay constant, and t is time in minutes. We will use an excel spread sheet to try to find an appropriate model. The model that I first tried was . I let a = 210 since this was the boiling point of my water. I found k using an excel spreadsheet and good old trial and error. You can see the y values that my model produced in the chart below along with the squares of the differences between the actual data and the values produced by my model.
Time 
Temperature 
y=210e^(.028*t) 
Square of the difference 




0 
210 
210.00 
0.00 
1 
200 
204.20 
17.65 
2 
189 
198.56 
91.46 
3 
180 
193.08 
171.10 
4 
174 
187.75 
189.04 
5 
167 
182.57 
242.28 
6 
161 
177.52 
273.05 
7 
156 
172.62 
276.31 
8 
151 
167.86 
284.13 
9 
148 
163.22 
231.69 
10 
145 
158.71 
188.09 
11 
142 
154.33 
152.08 
12 
139 
150.07 
122.56 
13 
136 
145.93 
98.55 
14 
134 
141.90 
62.38 
15 
131 
137.98 
48.72 
16 
128 
134.17 
38.07 
17 
126 
130.47 
19.94 
18 
125 
126.86 
3.47 
19 
123 
123.36 
0.13 
20 
121 
119.95 
1.09 
21 
119 
116.64 
5.56 
22 
117 
113.42 
12.81 
23 
116 
110.29 
32.61 
24 
114 
107.24 
45.64 
25 
113 
104.28 
75.99 
26 
112 
101.40 
112.29 
27 
110 
98.60 
129.88 
28 
109 
95.88 
172.11 
29 
108 
93.23 
218.05 
30 
107 
90.66 
267.02 
Sum of the squares 
3583.74918 

119.458306 
By calculating the difference between my model and the actual data using the least squares method we can see that our model needs to be improved on. 119.46 is much higher than we would like; we would prefer that the difference between the actual data and the model be much closer to zero.
Now lets look at it graphically.
We can see that our model does not fit the curve of the actual data well. Although our model is decreasing it is decreasing too slowly for the first 20 minutes and then too quickly during the last 10.
How can we improve this model? We must now consider that most decay models approach the value of zero but this is not the case for our model. This experiment was conducted in a room that was 77 degrees so once our water cools completely it should maintain a temperature of the air in the room or 77 degrees.
We must now consider Newton’s Law of Cooling. Newton’s law of cooling states that the final temperature of an object that is warmer than the air around it will be the room temperature. This temperature can be determined by the formula , where is the final temperature, is the initial temperature and is the room temperature.
The model we will use is . You can see from the sum of the squares of the difference that, at 34.72, we were much closer to zero this time. You can also see from the graph that our model appears to have the same general shape as our original data.
Time 
Temperature 
y =77+133e^(.0586t) 
Square of the Difference 




0 
210 
210.00 
0.00 
1 
200 
202.43 
5.91 
2 
189 
195.29 
39.58 
3 
180 
188.56 
73.25 
4 
174 
182.21 
67.39 
5 
167 
176.22 
85.03 
6 
161 
170.57 
91.66 
7 
156 
165.25 
85.52 
8 
151 
160.23 
85.10 
9 
148 
155.49 
56.07 
10 
145 
151.02 
36.25 
11 
142 
146.81 
23.12 
12 
139 
142.83 
14.71 
13 
136 
139.09 
9.53 
14 
134 
135.55 
2.41 
15 
131 
132.22 
1.49 
16 
128 
129.08 
1.16 
17 
126 
126.11 
0.01 
18 
125 
123.32 
2.83 
19 
123 
120.68 
5.37 
20 
121 
118.20 
7.86 
21 
119 
115.85 
9.91 
22 
117 
113.64 
11.29 
23 
116 
111.55 
19.76 
24 
114 
109.59 
19.46 
25 
113 
107.73 
27.74 
26 
112 
105.98 
36.19 
27 
110 
104.33 
32.10 
28 
109 
102.78 
38.70 
29 
108 
101.31 
44.74 
30 
107 
99.93 
50.02 
Sum of the squares 1041.616
34.72
Lets use our model to predict the temperature at 45, 60 and 300 minutes. Click here to see the model. Move point b to determine the temperature at any given time. Pay attention to the temperature as the time increases and record your results for 45 minutes, 60 minutes and 300 minutes.
Scroll Down For The Solutions
Solutions: 45 minutes  86 degrees, 60 minutes  81 degrees, and 300 minutes  77 degree
Did you notice that as the time increased
the temperature went to room temperature, 77 degrees. For further investigation
you might want to determine when the model first reached 77 degrees and what
this means. You might also want to investigate whether the model will drop
below 77 degrees and when this occurs.
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