Problem of the Week
The Problems - Bull's Eye!
02/14/11
In the game of darts, players take turns throwing darts at a circular target like the one shown. The dartboard, as it is called, is typically affixed to a wall and consists of a series of concentric circles that form rings which are then divided into 20 equally sized sections. Players are awarded points based on where the dart lands on the board when thrown. If the dart lands in the outer ring (colored green and red) a player is awarded two times the value that corresponds to that section. Similarly, if the dart lands in the inner ring (also colored green and red) a player is awarded three times the value that corresponds to that section. For this reason, these two rings are known as the Double Ring and Triple Ring, respectively. In the center of the dartboard is a red circle surrounded by a green ring. The green ring, which is commonly referred to as the Bull or Outer Bull’s Eye has a value of 25 points, while the red circle, the Bull’s Eye, is worth 50 points. If a player’s dart lands in one of the black or white areas of the scoring region he/she is simply awarded the point value associated with that section. A player is awarded no points if the dart lands anywhere on the perimeter of the board (beyond the outer edge of the Double Ring) or fails to stick to the board.
Four friends, Lucia, Marcus, Valerie and Kiefer will play a game of darts using a dartboard with the dimensions shown.
The distance from the center of the Bull’s Eye to the outer edge of the Double Ring is 16 cm and the ring itself is 1 cm in thickness.
The distance from the center of the Bull’s Eye to the outer edge of the Triple Ring is 11 cm and the ring itself is 1 cm in thickness.
The Bull has a radius of 16 mm and the Bull’s Eye has a radius of 6 mm.
The highest score a player can achieve in one throw is 60 points, which is earned by having the dart land in the Triple Ring section corresponding to the “20”. Valerie will go first. If Valerie’s dart lands within the scoring region of the dartboard (not beyond the outer edge of the Double Ring), what is the probability that she will score 60 points on her very first attempt? Express your answer as a common fraction.
well first off you have to eat a chicken then go to the bathroom and then eat your poop.
To find the probability that Valerie will score 60 points on her very first attempt, we need to consider the area of the scoring region that corresponds to 60 points.
From the information given, we know that the highest score a player can achieve in one throw is 60 points, which is earned by having the dart land in the Triple Ring section corresponding to the "20".
The radius of the Triple Ring is 11 cm. So, the area of the Triple Ring is π * (11^2) - π * (10^2) = 121π - 100π = 21π square cm.
Now, let's calculate the area of the scoring region for 60 points. Since each section of the dartboard is equally sized, the area of the scoring region for 60 points is (21π / 20) square cm.
Therefore, the probability that Valerie will score 60 points on her very first attempt is (21π / 20) / (π * 16^2) = (21 / 20) / (256) = 21 / (20 * 256) = 21 / 5120.
So, the probability that Valerie will score 60 points on her very first attempt is 21 / 5120, expressed as a common fraction.
To find the probability that Valerie will score 60 points on her first attempt, we need to determine the area of the region on the dartboard where she can score 60 points and divide it by the total area of the scoring region.
First, let's calculate the area of the region where Valerie can score 60 points. The only way to score 60 points is by landing the dart in the Triple Ring section corresponding to the '20'. The Triple Ring has a radius of 11 cm and is 1 cm thick. We need to find the area of the annulus (ring-shaped region) formed by the Triple Ring.
The area of an annulus can be found by subtracting the area of the inner circle from the area of the outer circle. The area of a circle is given by the formula A = π * r^2.
Area of the outer circle = π * (radius + thickness)^2 = π * (11 + 1)^2 = π * 12^2 = 144π
Area of the inner circle = π * radius^2 = π * 11^2 = 121π
Area of the annulus = Area of the outer circle - Area of the inner circle = 144π - 121π = 23π
Now let's calculate the total area of the scoring region. For this, we need to find the area of the entire dartboard excluding the perimeter (beyond the outer edge of the Double Ring). The radius of the dartboard is 16 cm.
Area of the entire dartboard = π * radius^2 = π * 16^2 = 256π
Finally, we can calculate the probability by dividing the area of the region where Valerie can score 60 points by the total area of the scoring region.
Probability = (Area of the annulus) / (Area of the entire dartboard) = (23π) / (256π) = 23/256
Therefore, the probability that Valerie will score 60 points on her very first attempt is 23/256.