2021 AMC 12A Fall Problem 9

Attempt Problem 9 of the 2021 AMC 12A Fall below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12A Fall solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

A right rectangular prism whose surface area and volume are numerically equal has edge lengths

\log_2 x,

\log_3 x,

and

\log_4 x.

What is

x?

2\sqrt{6}

6\sqrt{6}

24

48

576

With edges

a, b, c,

the equation

abc = 2(ab + bc + ca)

divides to

1 = 2\left(\tfrac{1}{a} + \tfrac{1}{b} + \tfrac{1}{c}\right)

\dfrac{1}{\log_2 x}

+ \dfrac{1}{\log_3 x}

+ \dfrac{1}{\log_4 x}

= \log_x 2 + \log_x 3 + \log_x 4

= \log_x 24

Let

a = \log_2 x,

b = \log_3 x,

c = \log_4 x.

Surface area equals volume gives

2(ab + bc + ca) = abc.

Dividing by

abc,

1 = 2\left(\frac{1}{c} + \frac{1}{a} + \frac{1}{b}\right).

Since

\dfrac{1}{a} = \log_x 2,

etc., the sum is

\log_x 2 + \log_x 3 + \log_x 4 = \log_x 24.

Thus

1 = 2\log_x 24,

\log_x 24 = \tfrac{1}{2},

meaning

x^{\frac{1}{2}} = 24

and

x = 576.

Thus, the correct answer is E.

Problem 9 in other years