Continuous dynamic systems49Figure 2.11.Example 2.14 Testing for art forgeries5All paintings contain small amounts of the radioactive element lead-210 and asmaller amount of radium-226. These elements are contained in white lead whichis a pigment used by artists. Because of the smelting process from which thepigment comes, lead-210 gets transferred to the pigment. On the other hand, over90 per cent of the radium is removed. The result of the smelting process is thatlead-210 loses its radioactivity very rapidly, having a half-life of about 22 years;radium-226 on the other hand has a half-life of 1,600 years (see example 2.7). Formost practical purposes we can treat radium-226 emissions as constant. Letl(t)denote the amount of lead-210 per gram of white lead at timet, andl0the amountpresent at the time of manufacture, which we take to be
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