$(30$ pts$)$ Water at $15C$ flows at speed $V$ around a cylindrical object of diameter $D$ as sketched. Angle $$ varies from $0$ to $180$ along the top of the cylinder, and the pressure along the top of the cylinder is approximated as $P=P_{\text{a}\text{t}\text{m}}+[\frac{1}{2}-2si{n}^2]V^2,$ where the local atmospheric pressure is $P_{\text{a}\text{t}\text{m}}=95\mathrm{.}8$kPa.

$($a$)$ If cavitation were to occur, at what angle $$ would it most likely occur?

$($b$)$ As the water speed $V$ increases, the pressure at the angle found in part $($a$)$ will decrease, and eventually it will equal the vapor pressure of the water, at which point cavitation will begin to occur. Using the given information, find the value of $V$ corresponding to the onset of cavitation.

$($c$)$ If the water temperature were to increase, would cavitation be more or less likely? In other words, would the speed corresponding to the onset of cavitation decrease or increase? Explain.