Bunuel wrote:

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

Attachment:

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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 =

4WIDTH = 2 radii =

2Rectangle's area = 4 * 2 = 8

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

Area of shaded regionThe "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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