$7\mathrm{.}50($See the news article "Ice Engineering" in Section $7\mathrm{.}9\mathrm{.}3.)$ A model study is to be done to determine the forces that a piece of ice in a river exerts on a bridge pier. The bridge pier in question has a rectangular cross$-$section. Assume that the force R is a function of the pier width b$,$ the ice thickness d$,$ the ice velocity V$,$ the acceleration due to gravity $8,$ and the ice density pi$,$ and the ice strength E$,$ which has dimensions FL$2.($a$)$ Based on these variables, determine the appropriate dimensionless variables for this problem. $($b$)$ The spherical state to be considered involves an ice thickness of $30$ cm and an ice velocity of $2$ m$/$s$.$ If the length scale is $1/10,$ what are the thickness and velocity of the model ice? $($c$)$ Explain whether the model ice can have the same strength as the spherical ice if the model ice and the spherical ice have the same density