The half-life of radium-226 is 1590 years. (a) A sample of radium-226 has a mass of 100 mg. Find a formula for the mass of the sample that remains after t years. (b) Find the mass after 1000 years correct to the nearest milligram. (c) When will the mass be reduced to 40 mg? SOLUTION (a) Let m(t) be the mass of radium-226 (in milligrams) that remains after t years. Then dm/dt = km and m(0) = 100, so this theorem gives m(t) = m(0)ekt = ekt. In order to determine the value of k, we use the fact that m(1590) = 1 2 . Thus e1590k = so e1590k = and 1590k = ln 1 2 = − ln(2) k = . Therefore m(t) = . We could use the fact that eln(2) = 2 to write the expression for m(t) in the alternative form m(t) = . (b) The mass after 1000 years is as follows. (Round your final answer to the nearest milligram.) m(1000) = ≈ mg