SMA ACFT ON A XCOUNTRY OUT OF YIP PERFORMS AN UNAUTH ENTRY INTO CLASS B AT DTW. THE PVT PF WAS NOT USED TO THE ACFT'S CLB PERFORMANCE AND OVERSHOT HER INTENDED ALT. INSTRUCTOR PLT WAS ON A FREQ CHANGE.

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MY STUDENT AND I WERE ON A XCOUNTRY IN A HIGH PERFORMANCE AIRPLANE FROM BTL-YIP-BTL. THERE WAS ALSO A STUDENT IN THE BACK READING CHKLISTS. I WAS HANDLING THE RADIOS. AFTER TAKING OFF FROM YIP WE PROCEEDED TO TURN W; ON COURSE. MYSELF AND THE STUDENT IN THE REAR SEAT INFORMED THE STUDENT FLYING TO STAY BELOW 3000 FT MSL. SHE LEVELED OFF JUST BELOW 3000 FT. I CALLED YIP TWR AND REQUESTED A FREQ CHANGE TO CONTACT DTW APCH. I CONTACTED DTW APCH AND RECEIVED A SQUAWK CODE. I THEN GLANCED OVER AT THE ALTIMETER AND IT READ 3500 FT AND CLBING. I INSTRUCTED THE STUDENT TO LOWER THE NOSE. I THEN TOOK THE ACFT AND LOWERED THE NOSE TO START A DSCNT. UPON DSNDING THROUGH 3500 FT I LOOKED AT THE GPS AND DETERMINED WE WERE CLR OF CLASS B AIRSPACE. SHORTLY THEREAFTER; THE DEP CTLR INFORMED US THAT WHEN CLBING OUT OF YIP; WE NEED TO STAY BELOW THE CLASS B UNTIL CLRED INTO IT. THE REASON FOR THIS AIRSPACE BUST IS MY TAKING MY ATTN AWAY FROM THE ALT FOR TOO LONG AND THE STUDENT NOT SETTING UP FOR LEVELOFF AND CRUISE FLT TO KEEP THE ACFT FROM CLBING. WITH CLB RATES OF 1000-2000 FPM IN THIS ACFT; IT DOES NOT TAKE LONG TO GET OFF OF AN ALT. STUDENTS ESPECIALLY; AND INSTRUCTORS ALSO; NEED TO MONITOR ALL PARAMETERS OF FLT CLOSELY WHEN OPERATING IN A CONGESTED AREA SUCH AS DTW. SUPPLEMENTAL INFO FROM ACN 381369: I DID NOT HAVE MUCH TIME IN THAT ACFT AND WASN'T USED TO THE CLB RATE.