logarithm applications

Question

polonium 210 has a half life of 140 days. Suppose a sample of the substance has a mass of 300 mg.

a) find the function that models the amount of the sample at time t.

b)find the mass remaining after one year.

c)how long will it take for the sample to decay to a mass of 200 mg?

Answer 1

Let A(t) = amount remaining in t days

 

A(t) = A₀e^kt, where A₀ is the initial amount and k is a constant

 

       = 300e^kt

 

Since the half life is 140 days, A(140) = 150 = 300e^140k

 

                                                              0.5 = e^140k

 

                                                            ln0.5 = 140k

 

                                                               k = -0.00495

 

So, A(t) = 300e^-0.00495t

 

b).  amount in 1 year = A(365) = 300e^{-0.00495(365)}

 

                                                = 300e^-1.80675

 

                                                = 49.2 mg

 

c).  Find t so that A(t) = 200:

 

      200 = 300e^-0.00495t

 

      0.6666666 = e^-0.00495t

 

      ln(0.666666) = -0.00495t

 

                   t = 81.9 days