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==Problem==
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#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 1 (Vieta's Formulas)==
 
 
 
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 = \boxed{(D) 30}</math>
 
 
 
- phuang1024
 
 
 
== Solution 2 (Rational Root Theorem bash) ==
 
 
 
We can find the roots of the cubic using the Rational Root Theorem, which tells us that the rational roots of the cubic must be in the form <math>\frac{p}{q}</math>, where <math>p</math> is a factor of the constant <math>(-6)</math> and <math>q</math> is a factor of the leading coefficient <math>(10)</math>. Therefore, <math>p</math> is <math>\pm (1, 2, 3, 6)</math> and q is <math>\pm (1, 2, 5, 10).</math>
 
 
 
Doing Synthetic Division, we find that <math>3</math> is a root of the cubic:
 
<cmath>\begin{array}{c|rrrr}&10&-39&29&-6\\3&&30&-27&6\\\hline\\&10&-9&2&0\\\end{array}.</cmath>
 
 
 
Then, we have a quadratic <math>10x^2-9x+2.</math> Using the Quadratic Formula, we can find the other two roots:
 
<cmath>x=\frac{9 \pm \sqrt{(-9)^2-4(10)(2)}}{2 \cdot 10},</cmath>
 
which simplifies to <math>x=\frac{1}{2}, \frac{2}{5}.</math>
 
 
 
To find the new volume, we add <math>2</math> to each of the roots we found:
 
<cmath>(3+2)\cdot(\frac{1}{2}+2)\cdot(\frac{2}{5}+2).</cmath>
 
Simplifying, we find that the new volume is <math>\boxed{30}.</math>
 
 
 
-MathWizard09
 
 
 
==Video Solution==
 
 
 
https://youtu.be/08YkinzFcCc
 
 
 
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
 
 
 
==See also==
 
{{AMC12 box|year=2022|ab=A|num-b=14|num-a=16}}
 
{{MAA Notice}}
 

Latest revision as of 00:36, 12 November 2022