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Elena is designing a paint can with thickness $t$ millimeters and height $h$ centimeters. She calculates that the thickness of the can in millimeters must be at least $0$.$1$ times the height of the can in centimeters in order to withstand pressure. Due to cost constraints, the cost of material used, $(0.2+t+0.5 h)$ cents, must be at most $12$.$2$ cents. Which of the following systems of inequalities best models the relationship between height and thickness described?$\newline$Choose $1$ answer:$\newline$(A) $\left\{\begin{array}{l}t \leq 12-\frac{h}{2} \\ 10 t \geq h\end{array}\right.$$\newline$(B) $\left\{\begin{array}{l}t \leq 12-\frac{h}{2} \\ t \geq 10 h\end{array}\right.$$\newline$(c) $\left\{\begin{array}{l}t \geq 12-\frac{h}{2} \\ 10 t \geq h\end{array}\right.$$\newline$(D) $\left\{\begin{array}{l}t \geq 12-\frac{h}{2} \\ t \geq 10 h\end{array}\right.$