2) (5 points) Suppose researchers have discovered a new radioactive isotope, we

Question

2) (5 points) Suppose researchers have discovered a new radioactive isotope, we shall call it Bryanium-23 (because I can). When trying to determine some basic properties of this new isotope, researchers observed its rate of decay. Two (2) weeks after its discovery, researchers had 45g of Bryanium-23. 3 weeks later, there was 28g. Suppose that Bryanium 23 decays continuously. Let A(t) describe the amount of Bryanium-23 A (in grams), given at time t (in weeks). Find an equation for A(t). Then find the number of weeks it will take for the isotope to decay to 3g

Explanation / Answer

2 weeks after discovery = 45 grams

5 weeks after discovery = 28 grams

Decay in one week = (28/45)1/3

= 0.62221/3

= 0.853

? One gram of isotope leaves 0.853 grams after one week

Let initial amount be Ao

Amount after 't' weeks = Ao(0.853)t

Ao x 0.8532  = 45

Ao = 45 x 1.3743

= 61.846

A(t) = Ao x (0.853)t

  = 61.846 x (0.853)t

A(n) = 3 grams

61.846 x (0.853)n = 3

0.853n = 3/61.846 = 0.0485

n = (log0.0485)/(log0.853)

= 19.033

? The isotope will decay to 3 grams in 19 weeks

? A(t) = 61.846 x (0.853)t

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