document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=109830>Question 109830</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A reboncing ball rebounces each time to a height equal to one half the height of the previous bounce. If it is dropped from a height of 16 mtrs. find the total distance it has travelled when it hits ground for the 10th time. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #80034 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=80034\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You can work this problem at least two ways. One way is to just do a table analysis of<BR>\n" );
document.write( "what happens ... as follows<BR>\n" );
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document.write( "Ball falls 16 feet and hits the floor first time. 16 feet ... 1 time<BR>\n" );
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document.write( "Next ball rebounds to 8 ft and falls back 8 ft where it hits floor second time. Total distance<BR>\n" );
document.write( "traveled to this point is 16 + 8 + 8 = 32 ft and 2 hits<BR>\n" );
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document.write( "Ball rebounds to 4 ft and falls back 4 ft where it hits 3rd time.  Total distance is 32 + 4 + 4<BR>\n" );
document.write( "= 40 ft and 3 hits<BR>\n" );
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document.write( "Ball rebounds to 2 ft and falls back 2 ft where it hits 4th time. Total distance now is<BR>\n" );
document.write( "40 + 2 + 2 = 44 ft and 4 hits.<BR>\n" );
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document.write( "Ball rebounds to 1 ft and falls back 1 ft where it hits 5th time. Total distance  is now<BR>\n" );
document.write( "44 + 1 + 1 = 46 ft and 5 hits<BR>\n" );
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document.write( "Ball rebounds to 1/2 ft and falls back 1/2 ft where it hits 6th time. Total distance up<BR>\n" );
document.write( "to this point is 46 + 1/2 + 1/2 = 47 ft and 6 hits.<BR>\n" );
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document.write( "Ball rebounds to 1/4 ft and falls back 1/4 ft to hit 7th time. Total distance now is<BR>\n" );
document.write( "47 + 1/4 + 1/4 = 47.5 ft and 7 hits<BR>\n" );
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document.write( "Ball rebounds to 1/8 ft (0.125 ft) and falls back 1/8 ft (again 0.125 ft) and hits for the<BR>\n" );
document.write( "8th time. Total distance now is 47.5 + 0.125 + 0.125 = 47.75 ft and 8 hits.<BR>\n" );
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document.write( "The ball rebounds to 1/16 ft (0.0625 ft) and falls back the same distance. The total distance<BR>\n" );
document.write( "is now 47.75 + 0.0625 + 0.0625 = 47.875 ft and 9 hits.<BR>\n" );
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document.write( "Finally the ball rebounds to 1/32 ft (0.03125 ft) and falls back the same amount, hitting <BR>\n" );
document.write( "the floor for the 10th and final time. Total distance = 47.875 + 0.03125 + 0.03125 = 47.9325 ft.<BR>\n" );
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document.write( "You could also do this problem using a geometric progression.<BR>\n" );
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document.write( "Suppose that instead of dropping the ball from a height of 16 ft to start the problem, that<BR>\n" );
document.write( "we shot it up 16 feet from floor level. The first cycle would be up 16 ft and down 16 ft for<BR>\n" );
document.write( "a total of 32 ft. The next cycle would be up 8 ft and down 8 ft for a total of 16 ft. Each <BR>\n" );
document.write( "cycle thereafter would be half of the preceding cycle. So in this progression the first<BR>\n" );
document.write( "term is 32 ft, the next 16, the next 8, and so on.<BR>\n" );
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document.write( "If you define a as the first term, r as the common ratio (in this problem r = 1/2), n<BR>\n" );
document.write( "as the number of terms to be considered (10 for this problem) and L as the last term, the<BR>\n" );
document.write( "equation for the sum (S) of the terms in a geometric progression is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28a+-+r%2AL%29%2F%281-r%29 ALIGN=MIDDLE  ALT=\"S+=+%28a+-+r%2AL%29%2F%281-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "in which <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+a%2Ar%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"L+=+a%2Ar%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "In this problem:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L+=+32%2A%281%2F2%29%5E%2810+-+1%29=+32%2A%281%2F2%29%5E9+=+32%2A%281%2F%282%5E9%29%29=+32%2A%281%2F512%29+=+0.0625 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now that we have L, we can return to the sum equation and substitute appropriate values:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28a+-+r%2AL%29%2F%281-r%29+=+%2832-%28%281%2F2%29%2A0.0625%29%29%2F%281-%281%2F2%29%29+=+%2832-0.03125%29%2F%281%2F2%29=+63.9375 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So this tells us that the total distance traveled is 63.9375 ft. But don't forget that<BR>\n" );
document.write( "in order to get this into the form of a geometric progression we added 16 feet on the<BR>\n" );
document.write( "front end so that every cycle would be complete. Therefore, we need to subtract that 16 feet<BR>\n" );
document.write( "off because the ball did not start at ground level. It was dropped from the height of<BR>\n" );
document.write( "16 feet. Subtracting 16 feet results in 63.9375 ft - 16 ft = 47.9375 ft for the total<BR>\n" );
document.write( "distance traveled during the cycles. This agrees with the answer we got the first way<BR>\n" );
document.write( "we did it ...<BR>\n" );
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document.write( "Hope this helps you to understand the problem and how it can be solved.<BR>\n" );
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