Answer
A) $p(v)=\frac{8}{v}$
B) $p=8$ pounds per square inch
Work Step by Step
Denote the pressure and volume of the gas in syringe as $p$ and $v$ respectively. Since the pressure is inversely proportional to the volume, we can mathematically describe the relationship as \begin{equation}
p= k\frac{1}{v},
\end{equation} where $k$ is a constant ofr proportionality that must be determined.
A) We know that the pressure $4$ pounds per square inch when the volume is $2$ cubic inches. Use this to find $k$. \begin{equation}
\begin{aligned}
4&=\frac{1}{2}k \\
4\cdot 2&= k\\
8&= k.
\end{aligned}
\end{equation} Hence, the pressure as a function of volume is
\begin{equation}
p(v)= \frac{8}{v}.
\end{equation} B) When $v= 1$, we get.
\begin{equation}
p(1)= \frac{8}{1}= 8.
\end{equation} The pressure of the gas in the syringe is $8$ pounds per square inch when the volume is $1$ cubic inch.
