Answer:
[tex]\dfrac{A_0}{1024}\; \sf mm[/tex]
Step-by-step explanation:
When a certain medicine enters the bloodstream, it gradually dilutes, decreasing exponentially with a half-life of 3 days. The initial amount of the medicine in the bloodstream is A₀ milliliters.
The amount of medicine A(t)) in the bloodstream after t days can be modelled by the half-life formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Half Life Formula}}\\\\A(t)=A_0\left(\dfrac{1}{2}\right)^{\dfrac{t}{t_{\frac12}}}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$A(t)$ is the quantity remaining.}\\\phantom{ww}\bullet\;\textsf{$A_0$ is the initial quantity.}\\ \phantom{ww}\bullet\;\textsf{$t$ is the time elapsed.}\\\phantom{ww}\bullet\;\textsf{$t_{\frac12}$ is the half-life of the substance.}\end{array}}[/tex]
In this case, A₀ is the initial quantity and t = 30.
Therefore:
[tex]A(30)=A_0\left(\dfrac{1}{2}\right)^{\dfrac{30}{3}}\\\\\\A(30)=A_0\left(\dfrac{1}{2}\right)^{10}\\\\\\A(30)=A_0\cdot \dfrac{1}{2^{10}}\\\\\\A(30)=\dfrac{A_0}{1024}\;\sf mm[/tex]
Therefore, the amount in the bloodstream 30 days later will be 1/1024th of the initial amount.
