Question

A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. Write your answer in the form \((a \sqrt{b}+c) / d\), where \(a, b, c,\) and \(d\) are positive integers.

Short answer

The answer is \((4\sqrt{1} + 0) / 16\)

Step-by-step solution

Visualizing and Identifying Components

Visualize a square and mark its center. From the center draw lines to the mid points of the sides of the square. This forms another smaller square inside the big square, such that any point inside this smaller square is closer to the center of the big square than to any of its sides. Let's assume the side of the big square be \(a\), then half its side would be \(a/2\), which would also be the sideness of the smaller square.

Calculate Areas

Calculate the area of the bigger square, which is \(Area_{Big} = a^2\). Similarly, calculate the area of the smaller square, which is \(Area_{Small} = (a/2)^2 = a^2/4\).

Compute Probability

Probability that the dart will hit nearer to the center than to any edge is equal to the ratio of the area of the smaller square to the area of the bigger square, which is \(Probability = Area_{Small} / Area_{Big} = (a^2/4) / a^2 = 1/4 \) which will simplify to \( (4\sqrt{1}+0) / 16 \)