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 $5\rightarrow \boxed{\text{E}}$.

See Also

1992 AJHSME (ProblemsAnswer KeyResources)
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
All AJHSME/AMC 8 Problems and Solutions

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