2010 AMC 8 Problem 18

Question

A decorative window is made up of a rectangle with semicircles on either end. The ratio of

AD

to

AB

is

3:2,

and

AB=30

inches. What is the ratio of the area of the rectangle to the combined areas of the semicircles?

\ 2:3

\ 3:2

\ 6:\pi

\ 9:\pi

\ 30 :\pi

Use the ratio

AD:AB=3:2

to express the rectangle height in terms of

AB

.

The two semicircles combine to one full circle.

Combining the semicircles would make a circle of diameter

d=30.

This would make the radius equal to

\dfrac{d}{2}.

Therefore, the combined area of the semicircles is

(\dfrac{d}{2})^2 \cdot \pi = \dfrac{\pi d^2}{4}.

Side

AD = \dfrac 32 d

because

AD:AB=3:2

and

AB=d.

The area of the rectangle is therefore

\dfrac 32 d^2.

The ratio of the area of the rectangle to the area of the semicircles is

\dfrac{\dfrac 32 d^2}{\dfrac{\pi d^2}{4}} = \dfrac{6}{\pi}.

Thus, the answer is C .

Problem 18 in other years