bonanova Posted July 11, 2009 Report Share Posted July 11, 2009 A circular [annular] race track has a constant width of w meters. If the longest straight-line distance entirely on the track is 12 meters, what is the area of the track? [Excluding the circular infield.]. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 11, 2009 Report Share Posted July 11, 2009 A = pir^2 - pi(r-w)^2 A = pir^2 - (pi(r^2-2rw+w^2)) A = pir^2 - pir^2 + 2pirw - piw^2 A = 2*pi*r*w - pi*w^2 Know: 2pir = 12 A = 12w - piw^2 and w <= 6/pi Quote Link to comment Share on other sites More sharing options...
0 plainglazed Posted July 11, 2009 Report Share Posted July 11, 2009 36pi? At least for one case and am working out the math Quote Link to comment Share on other sites More sharing options...
0 plainglazed Posted July 11, 2009 Report Share Posted July 11, 2009 (edited) no longer my strong point: R = radius of outer circle r = radius of the inner circle (R - w)2 + (12/2)2 = R2 R2 - 2RW + W2 + 36 = R2 R = w/2 + 18/w r = R - w = 18/w - w/2 pi(w/2 + 18/w)2 - pi(18/w - w/2)2 reduces to 36pi or generally for L = longest straight line distance on the track: (L/2)2pi Edited July 11, 2009 by plainglazed Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 11, 2009 Author Report Share Posted July 11, 2009 A = pir^2 - pi(r-w)^2 A = pir^2 - (pi(r^2-2rw+w^2)) A = pir^2 - pir^2 + 2pirw - piw^2 A = 2*pi*r*w - pi*w^2 Know: 2pir = 12 A = 12w - piw^2 and w <= 6/pi get rid of all those w's? You can if you ditch the idea that 2pir = 12 Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted July 11, 2009 Author Report Share Posted July 11, 2009 no longer my strong point: R = radius of outer circle r = radius of the inner circle (R - w)2 + (12/2)2 = R2 R2 - 2RW + W2 + 36 = R2 R = w/2 + 18/w r = R - w = 18/w - w/2 pi(w/2 + 18/w)2 - pi(18/w - w/2)2 reduces to 36pi or generally for L = longest straight line distance on the track: (L/2)2pi plainglazed has it. Can you do it without talking to Pythagoras? See spoiler in OP Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 11, 2009 Report Share Posted July 11, 2009 ok, not arguing in any way with the answer given by plainglazed, but I am having trouble conceptualizing, as classroom math was a long way away in reverse. I assume that the longest line possible in the OP is a tangent to the inner circle. If that is true, why is the then area not in any way dependant upon either the diameter of either of the circles or the value w, and actually is a constant under any set of variables? Quote Link to comment Share on other sites More sharing options...
0 plainglazed Posted July 11, 2009 Report Share Posted July 11, 2009 the general equation I gave at the end defines the area (A) relative to the longest line possible (L) as A=pi(L/2)2. The OP had given this variable to be 12 m. The elegant solution comes from defining the tracks inner circle radius as zero so the longest stright line (L) is the diameter of the outer circle. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 12, 2009 Report Share Posted July 12, 2009 the general equation I gave at the end defines the area (A) relative to the longest line possible (L) as A=pi(L/2)2. The OP had given this variable to be 12 m. The elegant solution comes from defining the tracks inner circle radius as zero so the longest stright line (L) is the diameter of the outer circle. That explanation finally makes it quite clear to me and I thank you. This OP is highly reminiscent of one Bonanova posted some time ago entitled 'hole in a sphere'. Mathematics, though my view of it is mostly nostalgic, forever presents itself as a recipe of equal parts of science and art. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted July 13, 2009 Report Share Posted July 13, 2009 get rid of all those w's? You can if you ditch the idea that 2pir = 12 I totally misinterpreted the longest straight line distance on the track was 12. I read it as "the longest path that can be taken on the track" <shrug> Quote Link to comment Share on other sites More sharing options...
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bonanova
A circular [annular] race track has a constant width of w meters.
If the longest straight-line distance entirely on the track is 12 meters,
what is the area of the track? [Excluding the circular infield.]
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