Thomas and Jonelle are playing darts in their garage using the board with the point values for each region shown below. The radius of the outside circle is 10 inches, and each of the other circles has a radius 2 inches smaller than the next larger circle. All of the circles have the same center. Thomas has only 1 dart left to throw and needs at least 30 points to win the game. Assuming that his last dart hits at a random point within a single region on the board, what is the percent chance that Thomas will win the game?
A. 36%
B. 30%
C. 16%
D. 9%
E. 1 1 / 2 %
36
how do you know that's the answer
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To calculate the percent chance that Thomas will win the game, we need to determine the probability of his last dart landing in a region that will give him at least 30 points.
Let's analyze the point values for each region:
- The outside circle has a radius of 10 inches, which means it has a diameter of 20 inches. This region has a point value of 5.
- The second circle has a radius of 8 inches (10 inches - 2 inches), which means it has a diameter of 16 inches. This region has a point value of 10.
- The third circle has a radius of 6 inches (8 inches - 2 inches), which means it has a diameter of 12 inches. This region has a point value of 20.
- The fourth circle has a radius of 4 inches (6 inches - 2 inches), which means it has a diameter of 8 inches. This region has a point value of 30.
- The fifth circle has a radius of 2 inches (4 inches - 2 inches), which means it has a diameter of 4 inches. This region has a point value of 40.
Since Thomas needs at least 30 points to win the game, he needs his last dart to land in the fourth or fifth circle.
To calculate the probability, we need to compare the area of the fourth and fifth circles with the total area of the board. We can find the area of a circle using the formula: A = πr^2, where A is the area and r is the radius.
Let's calculate the areas:
- Area of the fourth circle (30-point region): A4 = π(4^2) = 16π
- Area of the fifth circle (40-point region): A5 = π(2^2) = 4π
Considering that the area of the entire board is π(10^2) = 100π, the probability is the sum of the areas of the fourth and fifth circles divided by the total area of the board: (A4 + A5) / Total Area = (16π + 4π) / (100π) = 20π / 100π = 20/100 = 0.2 = 20%.
Therefore, the percent chance that Thomas will win the game is 20%.
None of the given answer options match 20%.