MAT2500-01/03 21S Quiz 1 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 EQUAL SIGNS and arrows 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). Only use technology to CHECK hand calculations, not subsitute for them.Given the two spheres described by the equations:
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
a) Complete the squares to find the coordinates of their centers and their radii: 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.
b) Evaluate the distance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn between the centers. [As a check, the fractional part of its numerical value is 0.74 to 2 decimal places.]c) Do the two spheres overlap or not (compare LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn with their radii)?d) Make a rough hand diagram of the cross-section plane through the symmetry axis connecting the centers of these spheres by plotting a circle of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjcjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== at the origin of the x-y plane and a circle of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEjcjJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJ0YyL0Y2USdub3JtYWxGJw== at a distance LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn along the positive x-axis. Include unit tickmarks on each axis. Be sure to label the points on the x-axis where the two circles intersect that axis with their numerical values to 2 decimal places when not integers. Does this diagram confirm your conclusion to part c)?
e) Derive the most simplified form of the equation for the plane of points which are equidistant from the two centers.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJkRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI+RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2JVExY2VudGVyc2VwYXJhdGlvbkYnRi9GMkY9RkA=The separation of the circles and therefore of the corresponding spheres is 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 the centers are farther apart than the sum of the radii, which means the spheres do not overlap. Rotating the plot shows this clearly.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If we want to plot the centers and connecting line segment, we just go a bit farther.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 the radii and separation distance we get this simple plot.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 pick the window to agree with the above 3d plot.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