The dartboard below consists of 5 regions bordered by concentric circles. The innermost region has a radius of r centimeters. Each of the 4 outer regions has uniform width r centimeters. Shown within each region is the number of points awarded (10, 8, 6, 4, or 1) to a player whose thrown dart sticks into that region. A player is awarded 0 points for any dart that does not stick into the board.In the dart game played on this dartboard, players take turns throwing 1 round consisting of 3 consecutive throws by 1 player. A player's 1-round point total is the sum of the point values awarded for the throws in that round. The first player to accumulate at least 75 points is the winner of the dart game; the winner need not complete his or her final round. Assume that a dart, when thrown, will stick into a random point on the dartboard. What is the probability that the dart will stick into the 10-point region?
Question
The dartboard below consists of 5 regions bordered by concentric circles. The innermost region has a radius of r centimeters. Each of the 4 outer regions has uniform width r centimeters. Shown within each region is the number of points awarded (10, 8, 6, 4, or 1) to a player whose thrown dart sticks into that region. A player is awarded 0 points for any dart that does not stick into the board.In the dart game played on this dartboard, players take turns throwing 1 round consisting of 3 consecutive throws by 1 player. A player's 1-round point total is the sum of the point values awarded for the throws in that round. The first player to accumulate at least 75 points is the winner of the dart game; the winner need not complete his or her final round. Assume that a dart, when thrown, will stick into a random point on the dartboard. What is the probability that the dart will stick into the 10-point region?
Solution
The probability of a dart sticking into the 10-point region is determined by the area of the 10-point region compared to the total area of the dartboard.
The 10-point region is the innermost circle with a radius of r. The area of a circle is given by the formula πr². So, the area of the 10-point region is πr².
The total area of the dartboard includes the 10-point region and the 4 outer regions, each with a width of r. This makes the total radius of the dartboard 5r (r for the 10-point region and 4r for the 4 outer regions). So, the total area of the dartboard is π(5r)² = 25πr².
The probability of a dart sticking into the 10-point region is therefore the area of the 10-point region divided by the total area of the dartboard, which is (πr²) / (25πr²) = 1/25 = 0.04 or 4%.
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