1992 AJHSME Problems/Problem 5

Problem

A circle of diameter $1$ is removed from a $2\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?

$[asy] fill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray); draw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1)); fill(circle((1,5/4),1/2),white); draw(circle((1,5/4),1/2),linewidth(1)); [/asy]$

$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

Solution

The area of the shaded region is the area of the circle subtracted from the area of the rectangle.

The diameter of the circle is $1$ , so the radius is $1/2$ and the area is $(1/2)^2\pi = \pi /4.$

The rectangle obviously has area $2\times 3= 6$ , so the area of the shaded region is $6-\pi / 4$ . This is closest to $5\rightarrow \boxed{\text{E}}$ .

1992 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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1992 AJHSME Problems/Problem 5

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