Question

Sodium - 24 is a radioactive isotope of sodium that is used to study circulatory dysfunction. Assuming that 3 micrograms of sodium - 24 are injected into a person the amount A in micrograms remaining in that person after t hours is given by the equation A=3e^-0.046t.(a) graph this equation(b) what amount of sodium -24 remains after 7 hours.(c) what is the half life of sodium -24 round your answer to two decimal places(d) in how many hours will the amount of sodium -24 be 1 microgram. round your answer to two decimal places

Answer 1

Given the amount A, in micrograms remaining in that person after t hours is given by the equation;

`A=3e^(-0.046t)`

(a) The graph of the equation is;

(b) After 7 hours;

`\begin{gathered} t=7 \\ A=3e^(-0.046t) \\ A=3e^(-0.046\left(7\right)) \\ A=3e^(-0.322) \\ A=2.17\mu g \end{gathered}`

(c) At half life of sodium -24;

`\begin{gathered} A=(3)/(2) \\ A=1.5\mu g \\ 1.5=3e^(-0.046t) \\ (1.5)/(3)=e^(-0.046t) \\ 0.5=e^(-0.046t) \\ \ln\mleft(0.5\mright)=\ln\mleft(e^(-0.046t)\mright) \\ -0.6931=-0.046t \\ t=-(0.6931)/(-0.046) \\ t=15.07h \end{gathered}`

(d) When the amount of sodium -24 is 1 microgram;

`\begin{gathered} 1=3e^(-0.046t) \\ (1)/(3)=e^(-0.046t) \\ 0.3333=e^(-0.046t) \\ \ln\mleft(0.3333\mright)=\ln\mleft(e^(-0.046t)\mright) \\ -1.0986=-0.046t \\ t=-(1.0986)/(-0.046) \\ t=23.88h \end{gathered}`