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?

$\ 2:3$

$\ 3:2$

$\ 6:\pi$

$\ 9:\pi$

$\ 30 :\pi$

Answer: C

Concepts:circle area rectangle area ratio

Difficulty rating: 1240

Solution:

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 .

2010 AMC 8 Problem 18

Answer: C

Difficulty rating: 1240

Solution:

Problem 18 in Other Years