$15\mathrm{.}7($From $[29,50]$ To develop strategies to manage water quality in polluted lakes, biologists must determine the depth of sediments and the rate of sedimentation. It has been determined that the depth of sediment $D(t)($in centimeters$)$ with respect to time $($in years before $1990)$ for Lake Coeur d'Alene, Idaho, can be determined by the equation

$D(t)=155(1-e^{-0\mathrm{.}0133t})$

$($a$)$ Find $D(20).$ What does this value represent biologically?

$($b$)$ Does $D(t)$ have a horizontal asymptote? If so$,$ give a biological interpretation of the horizontal asymptote.