Question

I need help with my homework question please it has two parts

Answer 1

`\begin{gathered} (A)\text{ }1.36\text{ }hr \\ (B)\text{ }1.36\text{ }mg \end{gathered}`

Step-by-step explanation

(A) To find the half-life of the drug, apply the exponential decay formula:

`y=x(1-r)^t`

where y = amount remaining

x = initial amount = 225 mg

r = decay rate = 40% = 0.4

t = time elapsed

Substituting the given values into the equation above:

`\begin{gathered} y=225(1-0.4)^t \\ y=225(0.6)^t \end{gathered}`

The half-life will occur when the amount of the drug is halved:

`\begin{gathered} (225)/(2) \\ 112.5mg \end{gathered}`

Substitute that for y and solve for t:

`\begin{gathered} 112.5=225(0.6)^t \\ 0.5=0.6^t \end{gathered}`

Convert the exponential equation into a logarithmic equation:

`\begin{gathered} \log0.5=\log0.6^t \\ \Rightarrow\log0.5=t\log0.6 \\ \Rightarrow t=(\log0.5)/(\log0.6) \\ t=1.36\text{ }hr \end{gathered}`

That is the half-life of the drug.

(B) To find the amount of the drug left after 10 hours, substitute 10 for t in the decay equation:

`\begin{gathered} y=225*0.6^(10) \\ y=1.36\text{ }mg \end{gathered}`

That is the answer.