Use the sum formula for an infinite geometric series. Rearrange the formula to solve for the common ratio, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»r«/mi»«/mstyle»«/math», given that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»S«/mi»«mo»=«/mo»«mfrac»«mn»9«/mn»«mn»4«/mn»«/mfrac»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»a«/mi»«mo»=«/mo»«mn»3«/mn»«/mrow»«/mstyle»«/math».

From the diagram, note the ball drops «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo».«/mo»«mn»0«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mrow»«/mstyle»«/math» before bouncing back up to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»80«/mn»«mo»%«/mo»«/mrow»«/mstyle»«/math» of its original height. As such, the height after the first bounce will be «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»0«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mn»0«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»8«/mn»«/mrow»«/mfenced»«mo»=«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math». To determine the total vertical distance travelled, consider everything after the first bounce as a geometric series with a common ratio of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»r«/mi»«mo»=«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»8«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»a«/mi»«mo»=«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math». Since the ball is bouncing up, and then dropping back down, the total distance it travels will be twice the sum of the series. And, the initial fall of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo».«/mo»«mn»0«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mrow»«/mstyle»«/math», which is not part of the series, must be included in the final distance calculation.

The total vertical distance traveled is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»d«/mi»«mo»=«/mo»«mn»2«/mn»«mi»S«/mi»«mo»+«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math».



The total distance the ball will have travelled is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»45«/mn»«mo»§#160;«/mo»«mi mathvariant=¨normal¨»m«/mi»«/mstyle»«/math».