What's the best angle? - April 29 - May 3, 1996
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I went to a San Diego Padres baseball game on Saturday night with my friends David, Bill, and Ihor. The people behind us were talking about the best places to sit. One of them announced that the best place to sit is where your lines of sight between home plate and first base meet at a 25-degree angle, and that there is one such seat for which this is true. The second person agreed about the 25 degrees, but felt that there are more places that you could sit.Who is right?
Annie says: Solutions
While this was sort of a contrived problem, there were some very good answers. Eighteen people got it right; 6 people got it right but didn't provide a general solution, and 35 people got it wrong.
A number of people correctly said that there were a whole bunch of seats, and described how you might find them (check out Rosina Pannone's solution, which is excellent), but didn't describe what the set of solutions looked like, namely a circle (or two circles, as it were). So folks with credit for the general solution included the circle, while those who are only right didn't get part. Some folks were right in saying that there is more than one seat, but didn't provide much explanation beyond that, so they didn't get any credit.
I like Amy's solution in that she pointed out what she learned by experimenting with the angles and the circles. She also described the resulting figure as a 'figure 8', which is really nice.
Following are highlights. The names of all the people who submitted correct solutions and most of the solutions are also available.
Rosina Pannone Grade 10 Cheshire High School, Cheshire, CT The man who said that there is more than one seat where the lines of sight between home plate and first base meet at a 25 degree angle is correct. In order to figure this out, I drew a diagram of a baseball diamond. I used the line between home plate and first base as the base of a triangle. Then, since the third angle had to equal 25 degrees, that left 155 degrees to split up between the two base angles. There are many different ways to split up 155 degrees between two angles, and each produces different vertices where a good seat is. I did five triangles, and arrived at five different seats, and there are even more. Therefore, there is more than one best seat.
Amy Forster Age 11, Grade 7, Home schooling Wilkins/Forster family Crooked Tree Point Cygnet, Tasmania, Australia. The second person, who said you could sit in more than one seat, is correct. Solution: I drew two points, Home plate and first base,on a piece of paper. I cut a piece of paper so that it was at a 25 degree angle. I lined the 25 degree angled piece of paper with the home plate and first base, and put points whereever it was 25 degrees.The points I drew formed two circles that intersected at home plate and first base, so you got a sort of figure of 8 of possible places to sit with a 25 degree angle between home plate and first base. Since spectators can't sit in the field the possible seats that give a 25 degree view are two arcs on opposite sides of the players within the sitting area (assuming spectators can sit 360 degrees round the players; otherwise there is only one arc of possible seating. From doing this problem I discovered there must be a relationship for circles where lines from each end of a chord to a point on the circumference of the circle make the same angle for any point on the circumference.
Mike Sue Grade: 10 School: Granada The second guy is right. There are many places where you get this 25 degrees and they are all on 2 circles with the radius of approximately 106.5 ft. The circles are centered on the perpendicular bisector of the line segment between home and first approx. 96.5 ft. toward the pitcher mound, and the other 96.5 feet away from the pitcher mound. Let Home to First be a line segment from H to F. Select a point V such that angle HVF = 25 degrees. Then construct a circumscribed circle which can be called O about triangle HVF. the measure of arc HF = 2 times the measure of angle HVF = 50. Let X be any point on arc HVF, such that X can't be H or F. The measurement of angle HXF = one half of the measurement of arc HF = 25. Thus the locus of vertices X forms the measurement of angle HXF = 25 = the arcHXF where H and F aren't included. To find the center of circle O: Assume the baseball diamond is a 90 ft square. Therefore the line segment HF = 90 ft. Construct a perpendicular bisector to HF intersecting H at M. This line passes through the center of circle o. The measurement of angle HOF = measurement of arc HF = 50. The measurement of angle HOM = one half the angle HOF = 25. HM = one half of HF = 45. R = 45/(sin 45) = 106.5 ft. OM = 45/(tan 45) = 96.5 ft. Arc HXF can be on either side of segment HF.
Ben Warfield Garfield High Home of the 1995 AL West Champion Seattle Mariners I hope you don't tell the deal about the good seats to the Mariners management: they're always looking for ways to make the extra buck (or 8 million), and since there are a very large number of such seats (defined by the set of vertices of a triangle with a fixed opposite side of 90 feet and a vertex angle of 25 degrees), it might get a little rough on those of us who actually buy tickets there regularly if they decided to put a premium on them. I can also say that your basic assumption, at least for the Kingdome, is a crock: you are much better off with outfield seats out in the open than with perfectly angled seats under the roof on the 200 level. If you're planning on seeing one of the Cleveland games, you'd better call ahead for tickets, they're selling out. Boston plays at Seattle on the 30 and 31st of May and the 1st and 2nd of June. I'll be in section 116, myself. The seats described are on the circumference of any circle constructed such that the circumference passes through home plate and first base and the central angle to the bases is 50 degrees. There are probably three sets of relevant circles, each consisting of a large number of cirles very close to each other (to adjust for the pitch of the bleachers), one for each deck, each circle having a radius of just over 106 feet (I'm not sure how far up the 300 level is). Having thought about it some, I am willing to concede that that is a pretty good measure of a seat, but I maintain that the circles that extend to the 200 level, while the angles to the bases may be great, do not provide a worthwhile baseball experience. I hope you're not a Twins fan: Randy Johnson should be pitching.
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