2022 AMC 12A Problem 15

Below is the professionally curated solution for Problem 15 of the 2022 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 12A solutions, or check the answer key.

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Concepts:Vieta’s Formulas volume

Difficulty rating: 1630

15.

The roots of the polynomial $10x^3-39x^2+29x-6$ 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?

$\dfrac{24}{5}$

$\dfrac{42}{5}$

$\dfrac{81}{5}$

$30$

$48$

Solution:

Let the roots be $r,s,t.$ By Vieta's formulas, $r+s+t=\dfrac{39}{10},$ $rs+rt+st=\dfrac{29}{10},$ and $rst=\dfrac{6}{10}=\dfrac35.$

The new volume is $(r+2)(s+2)(t+2)=rst+2(rs+rt+st)+4(r+s+t)+8=\frac35+\frac{58}{10}+\frac{156}{10}+8=30.$

Thus, the correct answer is D.

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