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| − | ==Problem==
| + | #redirect [[2022 AMC 10A Problems/Problem 16]] |
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| − | The roots of the polynomial <math>10x^3 - 39x^2 + 29x - 6</math> are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by 2
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| − | units. What is the volume of the new box?
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| − | ==Solution==
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| − | Let <math>a</math>, <math>b</math>, <math>c</math> be the three roots of the polynomial. The lenghtened prism's area is <math>V = (a+2)(b+2)(c+2) = abc+2ac+2ab+2bc+4a+4b+4c+8 = abc + 2(ab+ac+bc) + 4(a+b+c) + 8</math>.
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| − | By vieta's formulas, we know that:
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| − | <math>abc = \frac{-D}{A} = \frac{6}{10}</math>
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| − | <math>ab+ac+bc = \frac{C}{A} = \frac{29}{10}</math>
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| − | <math>a+b+c = \frac{-B}{A} = \frac{39}{10}</math>.
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| − | We can substitute these into the expression, obtaining <math>V = \frac{6}{10} + 2(\frac{29}{10}) + 4(\frac{39}{10}) + 8 = (D) 30</math>
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| − | - phuang1024
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| − | ==See also==
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| − | {{AMC12 box|year=2022|ab=A|num-b=14|num-a=16}}
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| − | {{MAA Notice}}
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