A gas having $\gamma=\frac{5}{2}$ and volume 360 c.c. is suddenly compressed to 90 c.c. If the initial…

The problem describes an adiabatic compression process, governed by the relation $PV^\gamma = \text{constant}$, where $\gamma = \frac{5}{2}$.

Given initial pressure $P_1 = P$, initial volume $V_1 = 360\text{ c.c.}$, and final volume $V_2 = 90\text{ c.c.}$, the adiabatic relation yields $P_1 V_1^\gamma = P_2 V_2^\gamma$.

Solving for final pressure, $P_2 = P_1 \left(\frac{V_1}{V_2}\right)^\gamma = P \left(\frac{360}{90}\right)^{5/2}$.

Since $\frac{360}{90} = 4$, and $4^{5/2} = (4^{1/2})^5 = 2^5 = 32$, it follows that $P_2 = 32P$.

This matches option D.

Final answer: $\boxed{\text{D}}$