2011 AMC 12A Problem 10

Question

A pair of standard

6

-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle’s circumference?

\dfrac{1}{36}

\dfrac{1}{12}

\dfrac{1}{6}

\dfrac{1}{4}

\dfrac{5}{18}

Answer: B

Concepts:dice (probability)circle inequality

Difficulty rating: 1370

Small hint:

With diameter

d,

the area is

\dfrac{\pi d^2}{4}

and the circumference is

\pi d

Big hint:

The inequality

\dfrac{\pi d^2}{4} \lt \pi d

reduces to

d \lt 4

Solution:

For diameter

d,

area

\lt

circumference means

\dfrac{\pi d^2}{4} \lt \pi d,

i.e.

d \lt 4.

Since

d \ge 2,

this needs a sum of

2

or

3.

A sum of

2

has probability

\dfrac{1}{36}

and a sum of

3