Consider a circle with radius 1. (a) What percentage of the...

Consider a circle with radius 1.
(a) What percentage of the points lies within $\frac{1}{2}$ unit of the center?
(b) Let $c$ be a constant with $0<c<1 .$ What percentage of the points lies within $c$ unit of the center?

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Key Concepts

Area of a Circle

The area of a circle is determined by the formula A = ?r², where r is the radius. This concept is fundamental in understanding how the space occupied by a circle changes with its radius, and it is directly applied when comparing subareas within the circle.

Scaling of Areas with Radius Squared

When comparing areas of circles with different radii, it is crucial to recognize that the area scales with the square of the radius. This means that if the radius is scaled by a factor, the area is scaled by the square of that factor, which explains the proportional relationship between the area of a subregion and the total area it is compared with.

Geometric Probability

Geometric probability involves finding the likelihood of an event by comparing areas (or volumes) when points are uniformly distributed over a region. In problems like this, the probability of a point lying within a specific subarea of a circle is found by taking the ratio of the subarea to the total area of the circle.

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