In the figure above, the circles touch each other and touch the sides of the rectangle at the lettered points shown. The radius of each circle is 1. Of the following, which is the best approximation to the area of the shaded region?

(A) 6
(B) 4
(C) 3
(D) 2
(E) 1


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The area of the shaded region equals
(Rectangle area) - (circles' area)/2

Rectangle area: L * W
The circles' radii completely span the length and width of the rectangle
Radius of one circle = 1

LENGTH = 4 radii = 4
WIDTH = 2 radii = 2
Rectangle's area = 4 * 2 = 8

Area of both circles, where r = 1: \(2(πr^2) = 2π\)

Area of shaded region
The "extra" area between the rectangle's edges and the circles' edges consists of 8 equal regions. See diagram.
Four of 8 are shaded = 1/2 of extra is shaded.

To get total extra area, subtract circles' area from rectangle's area.
To get half the total extra area (shaded region), subtract circles' area from rectangle's area and divide by 2.

Half of extra area = area of shaded region:

(rectangle area, R) - (circles' area, C) divided by 2:

\(\frac{(R - C)}{2}\)

\(\frac{(8 - 2π)}{2}\)

\(π\approx{3.14}\)

\(\frac{(8 - 6.28)}{2}= \frac{1.72}{2} = .86\)

\(0.86\approx{1}\)

Answer E
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In the depths of winter, I finally learned
that within me there lay an invincible summer.
-- Albert Camus, "Return to Tipasa"

We can see that the area of the shaded region is half the area of the region inside of the rectangle but outside of the two circles. Thus, the area of the shaded region is half the difference between the area of the rectangle and the total area of the two circles.

Area of the rectangle = 2 x 4 = 8

Area of a circle = π x 1^2 = π

Area of shaded region = ½ x (8 - 2π) = 4 - π

Since π is approximately 3.14, the area of the shaded region ≈ 4 - 3.14 = 0.86 ≈ 1.

Answer: E
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