Question

A circle has been drawn in the center of a square. A dot will be randomly placed inside the square. What is the probability that the dot will be placed in the purple section?

A) 11%

B) 20%

C) 38%

D) 40%

Answer 1

Answer: D) 40%

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Work Shown:

Area of purple rectangle (ignore circular cutout for now) = base*height = 6*12 = 72

I'm assuming the center of the circle is located on the right edge of the purple rectangle. If so, then we can subtract off half the circle's area from the purple rectangle area

Semicircle Area = (pi*r^2)/2 = (pi*3^2)/2 = 14.1371669

Area of purple region = (Rectangle Area) - (Semicircle Area)

Area of purple region = 72-14.1371669

Area of purple region = 57.8628331

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Now divide this area by 144 as this is the total area of the entire largest square (12 by 12 square) to get 57.8628331/144 = 0.401825 which is approximate. This rounds to 0.40 and converts to 40% which explains how I got choice D as the answer

Answer 2

That probability is approx. 40%. See below.

Explanation:

That probability is the area of the purple section divided by the total area of the square before coloring purple.

This area of the square is (12 cm)^2, or 144 cm^2.

Half the area of the circle is pi*(3 cm)^2, or 9*pi cm^2.

Thus, the purple area is (144 cm^2) / 2 - (9/2)*pi cm^2, or (72 - 9*pi/2) cm^2. This is approx. 57.86.

The ratio of this purple area to the full area of the square is:

57.86 cm^2

--------------------------------- = 0.4 (approximately)

144 cm^2