The dimensions of the box that minimize the cost are approximately \( x \approx 4.9324 \) cm and \( h \approx 22.1959 \) cm. The minimum cost is approximately \( 65.69 \) dollars.
Thus, the final answers are:
\[
\boxed{x \approx 4.9324} \quad \text{and} \quad \boxed{h \approx 22.1959} \quad \text{with minimum cost} \quad \boxed{65.69}
\]