Question

Rectangle $ABCD$ has sides $CD=3$ and $DA=5.$ A circle of radius $1$ is centered at $A,$ a circle of radius $2$ is centered at $B,$ and a circle of radius $3$ is centered at $C.$ Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

$3.5$

$4.0$

$4.5$

$5.0$

$5.5$

Solution:

The parts of the three circles inside the rectangle are quarter-circles with radii $1$ , $2$ , and $3$ . Their total area is $\dfrac{\pi}{4}+\pi+\dfrac{9\pi}{4}=\dfrac{7\pi}{2}.$

Using $\pi\approx\dfrac{22}{7}$ , this total is about $11$ . The rectangle has area $3\cdot5=15$ , so the desired area is about $15-11=4$ .

Thus, B is the correct answer.

2014 AMC 8 Problem 20

Solution:

Problem 20 in Other Years