MAT2500-03/04 10S Final Exam Print Name (Last, First) _________________________________|__
Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use arrows and equal signs when appropriate. Always simplify expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation). Indicate where technology is used and what type (Maple, GC). Use technology to check any integrals you set up.
1. Given the point 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, find the new coordinates, in each case stating the angles both in radians (exactly, using inverse trig functions) and in degrees (1 decimal place accuracy) and use proper identifying symbols for all coordinates: a) cylindrical coordinates. b) spherical coordinates. Support your work with two diagrams, one of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn plane and one of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcnpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn half plane, each including a reference triangle locating the point with respect to the axes with all three sides labeled by their lengths and both axes labeled by their coordinate labels and showing the relevant angles. Show clearly how you obtain values of your coordinates from these diagrams. Do the angles look right in these diagrams?
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a) Describe the region LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EiUkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRicvRjVRJ25vcm1hbEYn by giving the appropriate intervals of the polar coordinates over the region, and draw in the diagram a typical radial cross-section, labeling its endpoints by the values of the radial coordinate, shading in the region with equally spaced radial cross-sections. State the ranges of the two polar coordinates as inequalities.b) Use polar coordinates to evaluate the three integrals 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 and 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 by hand exactly. Evaluate the coordinates 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, 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 of the centroid of the region LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5exactly and numerically. Locate the centroid on the diagram. Does it seem right? Explain.
3. a) Check that 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 satisfies the condition that it admit a potential function, i.e., is a conservative vector field.
b) Find a potential function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for it.c) Use the potential to evaluate the line integral 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 over any curve from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMEYnRjRGNEY0LUkjbWlHRiQ2I1EhRidGNA== to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiM0YnRjRGNEY0RjQ=.d) By Green's theorem LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUklbXN1YkdGJDYlLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYjNiQtSSNtaUdGJDYlUSJDRicvJSdpdGFsaWNHRjwvRjNRJ2l0YWxpY0YnRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JJm1vdmVyR0YkNiUtRk02J1EiRkYnLyUlYm9sZEdGPEZQL0YzUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GLzYuUS0mUmlnaHRBcnJvdztGJy8lLHBsYWNlaG9sZGVyR0Y8RjIvRjZRJnVuc2V0RicvRjlGZG9GOi9GPkZkby9GQEZkby9GQkZkby9GREZkb0ZFRkgvRkRGPC1GLzYtUSkmbWlkZG90O0YnRjJGNUY4L0Y7RjdGPS9GQEY3RkFGQy9GRlEsMC4xNjY2NjY3ZW1GJy9GSUZhcC1GTTYlUSJkRidGUEZSLUZYNiUtRk02J1EickYnRmduRlBGaW5GW29GXm9Gam8tRi82LVEiPUYnRjJGNUY4Rl5wRj1GX3BGQUZDL0ZGUSwwLjI3Nzc3NzhlbUYnL0ZJRl9xRi4tRiw2JUYuLUYjNiQtRk02J1EiUkYnRmduRlBGaW5GW29GMkZULUYvNi1RIn5GJ0YyRjVGOEZecEY9Rl9wRkFGQ0ZFRkgtSSZtZnJhY0dGJDYoLUYjNiUtRi82LVErJlBhcnRpYWxEO0YnRjJGNUY4Rl5wRj1GX3BGQUZDRkVGSC1GLDYlLUZNNiVGZm5GUEZSLUYjNiQtSSNtbkdGJDYkUSIyRidGMkYyRlRGMi1GIzYmRmByRmhxLUZNNiVRInhGJ0ZQRlJGMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGZ3MvJSliZXZlbGxlZEdGNy1GLzYtUSgmbWludXM7RidGMkY1RjhGXnBGPUZfcEZBRkMvRkZRLDAuMjIyMjIyMmVtRicvRklGYHQtRlxyNigtRiM2JUZgci1GLDYlRmVyLUYjNiQtRmpyNiRGZHNGMkYyRlRGMi1GIzYmRmByRmhxLUZNNiVRInlGJ0ZQRlJGMkZic0Zlc0Zoc0Zqc0ZocS1GTTYlUSNkQUYnRlBGUkYy , the line integral around the triangle from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMEY0RjQtSSNtaUdGJDYjUSFGJ0Y0 to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMEYnRjRGNEY0RjQ= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiM0YnRjRGNEY0RjQ= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIwRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRidGMEY0RjRGNA== is zero. Check this by evaluating directly (not using the potential) the line integral over the three sides of this triangle with the counterclockwise direction, i.e., show that their sum is zero. Make a diagram showing your attack of the problem. [Note that your line integral over the hypotenuse should equal part c) as a check. You can also separately check the other two line integrals in this way.]solution (on-line)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjpGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ==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The plots are not meant to be done with this sophistication by students who have only elementary Maple skills.1. cylindrical and spherical coordinatesa)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: a bit less than 45 degrees past the vertical axis at 90 degrees.Below: a little more than 45 degrees beyond the horizontal axis at 90 degrees: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b)\316\270 is the same of course.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. polar coordinate 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conservative vector 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line integral of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY5 is the difference of values of the potential at the endpoint (+) and the initial point (-):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 bottom side: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 parameter interval for the line segment is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZOUSYwLjBlbUYnLUZQNiRRIjFGJ0Y5Rjk=.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triangle left side: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 parameter interval for the line segment is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5LUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZOUSYwLjBlbUYnLUZQNiRRIjBGJ0Y5Rjk=.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2JVElc3Vic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYtLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSI9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIwRidGQS1GPjYtUSJ+RidGQUZDRkZGSEZKRkxGTkZQL0ZTUSYwLjBlbUYnL0ZWRmluLUY+Ni1RIixGJ0ZBRkMvRkdGMUZIRkpGTEZORlBGaG4vRlZRLDAuMzMzMzMzM2VtRictRiw2JVEieUYnRi9GMkY9LUYsNiVRInRGJ0YvRjJGW28tRiw2JVEiRkYnRi9GMi1GNjYkLUYjNiVGOkZbb0Zhb0ZBRkEtRj42LVEiO0YnRkFGQ0Zeb0ZIRkpGTEZORlBGaG5GVUZlbi1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0Zpbi8lJmRlcHRoR0ZmcC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GYnA2JkZkcEZncEZpcC9GXHFRJWF1dG9GJy1GLDYlUSIlRidGL0YyLUY+Ni1RKSZtaWRkb3Q7RidGQUZDRkZGSEZKRkxGTkZQL0ZTUSwwLjE2NjY2NjdlbUYnL0ZWRmlxLUYsNiVRInJGJ0YvRjItRj42LVEiJ0YnRkFGQ0ZGRkhGSkZMRk5GUC9GU1EsMC4xMTExMTExZW1GJ0Zqbi1GNjYkLUYjNiNGZG9GQUZecEZlbi1JKG1zdWJzdXBHRiQ2Jy1GPjYtUSsmSW50ZWdyYWw7RidGQS9GRFEmdW5zZXRGJy9GR0Zecy9GSUYxL0ZLRl5zL0ZNRjEvRk9GXnMvRlFGXnNGaG5Gam4tRiM2Iy1GWDYkUSIzRidGQS1GIzYjRlcvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZaLyUvc3Vic2NyaXB0c2hpZnRHRlpGYnFGZW4tRj42LVEwJkRpZmZlcmVudGlhbEQ7RidGQUZdc0Zfcy9GSUZec0Zhcy9GTUZec0Zjc0Zkc0ZobkZqbkZkbw==triangle hypotenuse: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and 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pledgeWhen you have completed the exam, please read and sign the dr bob integrity pledge if it applies and hand in with your answer sheets as a cover page, with the Lastname, FirstName side face up: "During this examination, all work has been my own. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed this exam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."
Signature: Date: TTdSMApJNlJUQUJMRV9TQVZFLzE1MzA5MTIwNFgqJSlhbnl0aGluZ0c2Ii8lJ2Nvb3Jkc0clKmNhcnRlc2lhbkdbZ2whIyUhISEiIyIjLCYlInhHIiIjJSJ5RyEiIywmRipGLUYsRis2Ig==TTdSMApJNlJUQUJMRV9TQVZFLzE3NTIwNjA4NFgqJSlhbnl0aGluZ0c2Ii8lJ2Nvb3Jkc0clKmNhcnRlc2lhbkdbZ2whIyUhISEiIyIjJSJ0RyIiITYiTTdSMApJNlJUQUJMRV9TQVZFLzE3NTIwNjI2MFgqJSlhbnl0aGluZ0c2IjYjLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIjIiMiIiIiIiE2Ig==TTdSMApJNlJUQUJMRV9TQVZFLzE3NTIwNjMzMlgqJSlhbnl0aGluZ0c2Ii8lJ2Nvb3Jkc0clKmNhcnRlc2lhbkdbZ2whIyUhISEiIyIjLCQlInRHIiIjLCRGKiEiIzYiTTdSMApJNlJUQUJMRV9TQVZFLzE3NTIwNjU0MFgqJSlhbnl0aGluZ0c2Ii8lJ2Nvb3Jkc0clKmNhcnRlc2lhbkdbZ2whIyUhISEiIyIjIiIhJSJ0RzYiTTdSMApJNlJUQUJMRV9TQVZFLzE3NTIyMjg3NlgqJSlhbnl0aGluZ0c2IjYjLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIjIiMiIiEiIiI2Ig==TTdSMApJNlJUQUJMRV9TQVZFLzE3NTIzOTUxMlgqJSlhbnl0aGluZ0c2Ii8lJ2Nvb3Jkc0clKmNhcnRlc2lhbkdbZ2whIyUhISEiIyIjLCQlInRHISIjLCRGKiIiIzYiTTdSMApJNlJUQUJMRV9TQVZFLzE3NTIzOTY4OFgqJSlhbnl0aGluZ0c2Ii8lJ2Nvb3Jkc0clKmNhcnRlc2lhbkdbZ2whIyUhISEiIyIjJSJ0RywmIiIkIiIiRikjISIkIiIjNiI=TTdSMApJNlJUQUJMRV9TQVZFLzE3NTI1NjE2MFgqJSlhbnl0aGluZ0c2IjYjLyUnY29vcmRzRyUqY2FydGVzaWFuR1tnbCEjJSEhISIjIiMiIiIjISIkIiIjNiI=TTdSMApJNlJUQUJMRV9TQVZFLzE3NTI1NjMzNlgqJSlhbnl0aGluZ0c2Ii8lJ2Nvb3Jkc0clKmNhcnRlc2lhbkdbZ2whIyUhISEiIyIjLCYlInRHIiImISInIiIiLCZGKiEiJiIiJ0YtNiI=