Difference between revisions of "2022 AMC 12A Problems/Problem 15"

Latest revision as of 00:36, 12 November 2022

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-==Problem==
+#redirect [[2022 AMC 10A Problems/Problem 16]]
-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
-units. What is the volume of the new box?
-==Solution==
-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>.
-By vieta's formulas, we know that:
-<math>abc = \frac{-D}{A} = \frac{6}{10}</math>
-<math>ab+ac+bc = \frac{C}{A} = \frac{29}{10}</math>
-<math>a+b+c = \frac{-B}{A} = \frac{39}{10}</math>.
-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>
-- phuang1024