SOLUTION: A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region that is the boundary of the region. A. 1/4

Question 1183924: A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region that is the boundary of the region.
A. 1/4
B. 1/3
C. 1/2
D. 2/3
E. 3/4
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849) (Show Source): You can put this solution on YOUR website!

Let be radius of the circle
Distance of midpoint is
Now Area of bigger circle,
Area of circle (area closer to center) with as radius

Area of region closer to circumference

Now, required probability
......simplify
Thus,

answer: B.

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region.

In the given circular region, draw the concentric circle that has half the radius of the given circle. The points inside the smaller circle are closer to the center of the circle than to the boundary of the circular region; the points in the circular region outside the smaller circle are closer to the boundary.

The smaller circle, with half the radius of the larger circle, has an area that is one-fourth the area of the whole region. So the probability that a random point in the circular region is closer to the center than to the boundary is 1/4.

ANSWER A: 1/4

The other tutor came up with an answer of 1/3, by comparing the area of the smaller circle to the area of the circular region outside the smaller circle. That calculation shows that the ODDS of having a random point in the circular region being closer to the center than to the boundary are 1:3.

But the problem asks for the probability -- not the odds.

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