2006 AMC 8 Problem 7

Question

Circle

X

has a radius of

\pi.

Circle

Y

has a circumference of

8 \pi.

Circle

Z

has an area of

9 \pi.

List the circles in order from smallest to largest radius.

X,

Y,

Z

Z,

X,

Y

Y,

X,

Z

Z,

Y,

X

X,

Z,

Y

Convert all three pieces of information into radii.

Use

C=2\pi r

for circle

Y

and

A=\pi r^2

for circle

Z

.

Recall that

C = 2 \pi r

and

A = \pi r^2.

Using these formulas we get that the radius of

Y

is

8 \pi \div (2 \pi) = 4.

We also get that the radius of

Z

is

\sqrt{9 \pi \div \pi} = 3.

As

\pi

is greater than

3

and less than

4,

the correct order is

Z,

X,

Y.

Thus, the answer is B .

Problem 7 in other years