Find an integer n such that Zn has

The amount of caffeine in the body after consumption follows an exponential decay model. After taking your coffee, 50% of caffeine is left in your body in about 6 hours. If you had 1 cup of coffee 9 hours ago how much caffeine is left in your system?

Show that the argument form with premises ${\displaystyle (p\wedge t)\Rightarrow (r\vee s),q\Rightarrow (u\wedge t),u\Rightarrow p,}$ and ${\displaystyle urc\text{or}\ne rs}$ and conclusion ${\displaystyle q\Rightarrow r}$ is valid by using rules of inferences.
$\begin{array}{|ccc|}\hline \text{Rule of Inference}& \text{Tautology (Deduction Theorem)}& \text{Name}\\ P& P\Rightarrow (P\vee Q)& \text{Addition}\\ \overline{\therefore P\vee Q}& & \\ P\wedge Q& (P\wedge Q)\Rightarrow P& \text{Simplification}\\ \therefore P& & \\ P& \left[\right(P)\wedge (Q\left)\right]\Rightarrow (P\wedge Q)& \text{Conjunction}\\ Q& & \\ \therefore \overline{P\wedge Q}& & \\ P& \left[\right(P)\wedge (P\Rightarrow Q\left)\right]\Rightarrow (P\wedge Q)& \text{Modus Ponens}\\ P\Rightarrow Q& & \\ \overline{\therefore Q}& & \\ ⌝Q& \left[\right(⌝Q)\wedge (P\Rightarrow Q\left)\right]\Rightarrow ⌝P& \text{Modus Tollens}\\ P\Rightarrow Q& & \\ \overline{\therefore ⌝P}& & \\ P\Rightarrow Q& \left[\right(P\Rightarrow Q)\wedge (Q\Rightarrow R\left)\right]\Rightarrow (P\Rightarrow R)& \text{Hypothetical Syllogism (''chaining'')}\\ Q\Rightarrow R& & \\ \overline{\therefore \Rightarrow R}& & \\ P\vee Q& \left[\right(P\vee Q)\wedge (⌝P\left)\right]\Rightarrow Q& \text{Disjunctive syllogism}\\ ⌝P& & \\ \overline{\therefore Q}& & \\ P\vee Q& \left[\right(P\vee Q)\wedge (⌝P\vee R\left)\right]\Rightarrow (Q\vee R)& \text{Resolution}\\ ⌝P\vee R& & \\ \overline{\therefore Q\vee R}& & \\ \hline\end{array}$

simplify (sinx/(1+sinx)) + ((1+sinx)/cosx)

Explain the difference between a first and a second- order linear differential equation

4𝑦 ′′(𝑡) + 24𝑦 ′ (𝑡) + 37𝑦(𝑡) = 0  𝑦(𝜋) = 1, 𝑦 ′ (𝜋) = 0

How many arrangements of the letters ABCDEFGH contain BHFE?

Use Laplace transform to solve the following system of equations x' = 7x - 4y, x(0)= 1 y' = 8x - 5y, y(0)=3

½+1/4

Fint dw/dt using the chain rule 

W=x²y³; x=t³ ;y=t²