A slice of pizza that is all crust

Question

A circular pizza has radius 12 cm. The crust is the outer ring from radius 9 cm out to 12 cm. A topping fleck sits at a uniformly random point of the pizza. What is the probability the fleck is on the crust?

Accepted Answer

The inner (non-crust) disk has radius 9 cm. Its share of the area is \left(\frac{9}{12}ight)^2 = \left(\frac34ight)^2 = \frac{9}{16} . The crust is everything else: P(	ext{crust}) = 1 - \frac{9}{16} = \frac{7}{16} = 0.4375 . Takeaway. An outer ring's probability is 1 - \left(	frac{r{	ext{in}}}{R}ight)^2. A crust only a quarter of the radius deep still holds nearly half the area, since the outer band is the widest part of the disk.

A slice of pizza that is all crust

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