8. The radioactive element Polonium- 210 has a relatively short half-life of 140 days, and one way to model the ing after t days is with the func- tion A(t) = A,e is the mass at time t = 0 (note that amount of Polonium-210 remain- where A -0.004951t %3D A, A(140) = 0.) What percentage of %3D the original mass of a sample of Polonium-210 remains after one year? in x -int s woNe gaph:LF] 3 96 リint

5. Differentiate the following natural log and e functions. a. b. C. y' = y' = y' 11 y" = y = In(12 - x - 5x²1) y = In(e-x + xe-x) y = x² In (8x)

4. Radioactive Decay of Carbon-14 (¹4C) The half-life of ¹4C is 5730 years (yr). If the original amount of ¹4C in a particular living organism is 20 grams (g) and found in a fossil of that organism 0.01 g, determine the approximate age of the fossil. Evaluate the remaining amount of ¹4C after 90,000 yr. Hints: Let m(t) represent amount of 14C at time t. We have the initial amount m(0) 20 g. The half-life time is T = - 5730 yr. a piece of information The 14C found in the fossil at an unknown time t* is m(t*) = 0.01 g We want to evaluate the approximate age of the fossil, i.e., we need to find the time t* at which the amount of ¹4C dropped from 20 g to 0.01 g. • The IVP that represents the decay model is dm dt m(0) = 20. (i) Solve the IVP, (ii) find k, (iii) the age of the fossil t and (iv) evaluate m(90000). = km,

7. A radioactive material, Whatsuppium (Wu) decays according to the equation W (t) = 120(1.21)-t/5, where W(t) is the amount remaining (in grams) after t years. What is the rate of change of Wu after 20 years? %3D Ans: