MAT2500-01/02 20S 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.1. a) Find an equation of a sphere in standard form if one of its diameters has endpoints LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNiktSSNtbkdGJDYlUSI1RicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSIsRidGNEY3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEiNEYnRjRGN0Y6LUYxNiVRIjNGJ0Y0RjdGNEY3RjRGN0Y0Rjc= and 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.
b) What are the center point and radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIxRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ== of this sphere?c) Find the center point and radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjVGQQ== of the second sphere: 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.d) What is the distance s between the centers? Evaluate numerically 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 and the ratios 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 .e) Based on d), do the two spheres overlap? Explain why, using the previous numerical values. [An optional sketch with circles illustrating these numbers could be useful.]solutionThe radius of the sphere with a diameter connecting them is half the distance between tthe two diameter endpoints, while the center position vector is the vector average.
We need display, circle, sphere commands and CompleteSquare from these packages: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 diameter endpoints and center of sphere 1 (average of the coordinates of the endpoints for the midpoint of the line segment connecting them), and the radius vector from the center to the point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjRDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRic= are 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 radius is the length of the "radiusvector" line segment (which points from the center to the point D1), namely the square root of the sum of squares of this 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The first sphere equation is 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The second sphere equation is given 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Complete the squares to discover the center and radius of the second sphere.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So center and radius are: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So we can plot these spheres and their ccnters using advanced knowledge of Maple.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVElcGx0MEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GLDYtUSomY29sb25lcTtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjc3Nzc3OGVtRicvRkVGUy1GRzYlUShkaXNwbGF5RidGSkZNLUkobWZlbmNlZEdGJDYkLUYjNi4tRkc2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGQy8lJmRlcHRoR0Zfby8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GW282JkZdb0Zgb0Ziby9GZW9RJWF1dG9GJy1GRzYlUSxwb2ludHBsb3QzZEYnRkpGTS1GWTYkLUYjNjAtRlk2Ji1GIzYmLUZHNiVRKENlbnRlcjFGJ0ZKRk0tRiw2LVEiLEYnRi9GMi9GNkZMRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnLUZHNiVRKENlbnRlcjJGJ0ZKRk1GL0YvLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmlwLUZHNiVRJmNvbG9yRidGSkZNLUYsNi1RIj1GJ0YvRjJGNUY3RjlGO0Y9Rj9GUkZULUZHNiVRJmdyZWVuRidGSkZNRmlwLUZHNiVRJ3N5bWJvbEYnRkpGTUZbci1GRzYlUSxzb2xpZHNwaGVyZUYnRkpGTUZpcC1GRzYlUStzeW1ib2xzaXplRidGSkZNRltyLUkjbW5HRiQ2JFEjMjBGJ0YvRi9GL0ZpcEZqbkYrRlUtRlk2JC1GIzYxLUZHNiVRJ3NwaGVyZUYnRkpGTS1GWTYkLUYjNiZGZnBGaXAtRkc2JVEocmFkaXVzMUYnRkpGTUYvRi9GaXBGYnMtRlk2JC1GIzYmRl9xRmlwLUZHNiVRKHJhZGl1czJGJ0ZKRk1GL0YvRmlwLUZHNiVRLXRyYW5zcGFyZW5jeUYnRkpGTUZbci1GW3M2JFElMC43NUYnRi9GaXAtRkc2JVEoc2NhbGluZ0YnRkpGTUZbci1GRzYlUSxjb25zdHJhaW5lZEYnRkpGTS8lK2V4ZWN1dGFibGVHRjRGL0YvRl91Ri9GL0ZfdUYvThe separation vector between the two centers 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Since the separation distance is smaller than both radii, the spheres must not only overlap but each center must lie inside the other sphere.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