Two children Qosay and Karam are playing a game in which they try to hit a small box on the floor with a marble fired from a spring$-$loaded gun that is mounted on a table. The target box is at a horizontal distance $D=2\mathrm{.}20m$ from the edge of the table $($See the figure$).$ Qosay compresses the spring $1\mathrm{.}10cm,$ but the center of the marble falls $27\mathrm{.}0cm$ short of the center of the box $($blue colored path$).($a$)$ How far should Karam compress the spring to score a direct hit? Assume that neither the spring nor the ball encounters friction in the gun?

$($b$)$ Assuming the marble $(A)$ and the box $(B)$ strek together after collision and enters into a region where the coefficient of kinctic frctuon. $i_8=0\mathrm{.}10$ and comes to a stop in distance $y$ within that region. What is the value of distance yes