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Slope Fields

Part 3: Slope Fields for Radioactive Decay

Next we construct a slope field for radioactive decay. Recall that the differential equation is the same as that for natural growth,

dP/dt = k P,

but now k is a negative constant.

Here again are the data from the Radioactive Decay module, as cited from a 1905 paper by Meyer and von Schweidler.

Time (days) Relative Activity
0.2 35.0
2.2 25.0
4.0 22.1
5.0 17.9
6.0 16.8
8.0 13.7
11.0 12.4
12.0 10.3
15.0 7.5
18.0 4.9
26.0 4.0
33.0 2.4
39.0 1.4
45.0 1.1

Source: S. Meyer and E. von Schweidler, Sitzungberichte der Akademie der Wissenschaften zu Wien, Mathematisch-Naturwissenschaftliche Classe, p. 1202 (Table 5), 1905.

Recall that we found a reasonable fit to these data with the formula P = P0bt, where P0 = 28 and b = 0.92466. (Your numbers may have differed slightly from these.)

  1. Rewrite the formula P = P0bt in the form P = P0ekt, where k is a negative constant.

  2. What differential equation of the form dP/dt = k P is satisfied by the exponential function in step 1? Draw a slope field for this differential equation. Add the exponential decay curve to your slope field.

  3. Overlay the data points on the picture in step 2. What does the slope field add to your understanding of the exponential model for radioactive decay?
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Last modified: October 7, 1997