2022 AMC 12A Problem 15

Question

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

Answer: D

Concepts:Vieta’s Formulas volume

Difficulty rating: 1630

Small hint:

By Vieta,

r+s+t=\dfrac{39}{10},

rs+rt+st=\dfrac{29}{10},

rst=\dfrac35

Big hint:

Expand

(r+2)(s+2)(t+2)

using these symmetric sums

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

\begin{gathered} (r+2)(s+2)(t+2) \\ =rst+2(rs+rt+st) \\ \quad {}+4(r+s+t)+8 \\ =\frac35+\frac{58}{10} \\ \quad {}+\frac{156}{10}+8 \\ =30. \end{gathered}

Thus, the correct answer is D.