Stewart Calculus 8e. 14.7.53.problem statementA cardboard box without a lid is to have a volume of 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. Find the dimensions that minimize the amount of cardboard used. (Let x, y, and z be the dimensions of the cardboard box.)
[4000 is a randomized number, so we will work this for an arbitrary volume V and at the end plug in the number.]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 some reason Maple won't solve 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 by just taking the third root because of complex number problems but "allvalues" forces it to push through. Annoying, but this we can actually solve by looking at the equations. Solve the first for y, substitute in the second to get a simple equation for x, solve, backsubstitute.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnYXNzdW1lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEiVkYnRi9GMi1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC1JI21uR0YkNiRRIjBGJ0ZBLyUrZXhlY3V0YWJsZUdGRUZBRkFGZW5GQQ==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 is the step by step solution (ignore complex roots).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backsub to get y: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evaluate z (not required by problem statement but it makes sense to report the dimensions!)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 evaluate the extremal area.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYsNiVRJXN1YnNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Mi1GLDYlUSJ4RidGNEY3LUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSi8lKXN0cmV0Y2h5R0ZKLyUqc3ltbWV0cmljR0ZKLyUobGFyZ2VvcEdGSi8lLm1vdmFibGVsaW1pdHNHRkovJSdhY2NlbnRHRkovJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZZLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnRkYtSSZtZnJhY0dGJDYoLUZqbjYkUSIxRidGRi1Gam42JFEiM0YnRkYvJS5saW5ldGhpY2tuZXNzR0Ziby8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zqby8lKWJldmVsbGVkR0Y2LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GQzYtUTEmSW52aXNpYmxlVGltZXM7RidGRkZIRktGTUZPRlFGU0ZVL0ZYUSYwLjBlbUYnL0ZlbkZmcC1GZ242JS1GLDYlUSNWfGlyRidGNEY3LUYjNiVGXW9GK0ZGRl9wLUZDNi1RIixGJ0ZGRkgvRkxGNkZNRk9GUUZTRlVGZXAvRmVuUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0Y0RjctRkM2LVEpJmVxdWFscztGJ0ZGRkhGS0ZNRk9GUUZTRlVGV0ZaRmZuRmJwRmhwRl9xLUZDNi1RIn5GJ0ZGRkhGS0ZNRk9GUUZTRlVGZXBGZ3AtRiw2JVEiQUYnRjRGNy1GOzYkLUYjNiZGP0ZfcUZlcUZGRkZGRkZGRkZGKy8lK2V4ZWN1dGFibGVHRkpGRg==The box has dimensions 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and the minimal area is 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
Since the area is a positive function on the interior of the first quadrant and infinite on the axis boundary, and has exactly one local extrema, it has to be a local and global minimum.The randomized number was chosen to eliminate the radical since 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 and: 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Even though the problem did not ask for the minimal area, clearly this is of interest to evaluate to complete the problem!