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How much of a radioactive substance?

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How much of a radioactive substance must be presented to decay to 60 grams in 11 years if the half-life of the substance is 3.9 years?

Select one:

a. Between 400 and 420 grams

b. Between 420 and 440 grams

c. Between 440 and 460 grams

d. Over 460 grams

Feb 21, 2019

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How much of a radioactive substance must be presented to decay to 60 grams in 11 years if the half-life of the substance is 3.9 years?

$$\begin{array}{rcll} A(t)&=&A_0\cdot \left(\dfrac12 \right)^\dfrac{t}{t_{1/2}},\ \text{where} \\ \\ A(t) &-& \text{the amount left after t years;} \\ A_0 &-& \text{the initial quantity of the substance that will undergo decay;} \\ t_{1/2} &-& \text{the half-life of the decaying quantity.} \\ \end{array}$$

$$\begin{array}{|rcll|} \hline A(t) &=& 60g \\ t_{1/2} &=& 3.9 y \\ t &=& 11y \\\\ \hline A(t)&=& A_0\cdot \left(\dfrac12 \right)^\dfrac{t}{t_{1/2}} \\ 60g &=& A_0\cdot \left(\dfrac12 \right)^\dfrac{11y}{3.9 y} \\ 60g &=& A_0\cdot \left(\dfrac12 \right)^\dfrac{11 }{3.9 } \\ 60g &=& A_0\cdot \left(\dfrac12 \right)^{2.82051282051} \\ 60g &=& A_0\cdot 0.5^{2.82051282051} \\ 60g &=& A_0\cdot 0.14156015773 \\\\ A_0 &=& \dfrac{60g}{0.14156015773} \\\\ \mathbf{A_0} & \mathbf{=} & \mathbf{423.848072521g} \\ \hline \end{array}$$

b. Between 420 and 440 grams

Feb 21, 2019