To win the carnival game Ring Ding, you must toss a wooden ring onto a grid of rectangles so that it lands without touching any of the grid lines. The ring has a 3-inch diameter, the rectangles are twice as long as they are wide, and the game has been designed so that you have a 28% chance of winning. What are the dimensions of each rectangle

Question

Mathematics

Hailee Gill

31 December, 01:35

To win the carnival game Ring Ding, you must toss a wooden ring onto a grid of rectangles so that it lands without touching any of the grid lines. The ring has a 3-inch diameter, the rectangles are twice as long as they are wide, and the game has been designed so that you have a 28% chance of winning. What are the dimensions of each rectangle

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Answers (1)

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Brisa Lawrence · Answer 1

5in x 10in

Step-by-step explanation:

Solution:

1. You have to assume that the ring will land on the board and as a result, just focus on one rectangle of the board since they are all the same size.

2. For the ring to land entirely inside the rectangle, the center of the ring needs to be 1.5 inches inside the border (radius is 1.5 inches), so the "winning rectangle" will have dimensions:

L = (x - 2 (1.5)) = x - 3

w = (2x - 2 (1.5)) = 2x-3.

3. Now, we will set up and solve our geometric probability equation. The winning rectangle's area of (x-3) * (2x-3) divided by the total rectangle's area of x * (2x) = 28%

(2x-3) * (x-3) / (2x^2) =.28

2x^2 - 9x + 9 =.56x^2

1.44x^2 - 9x + 9 = 0

4. Now you can plug that in the quadratic formula and get the solutions x = 1.25 or 5. However, it cannot by 1.25 because that is too small of a dimension for the the circle with diameter of 3 to fit in, the answers has to be 5.

5. As a result, the dimensions of the rectangles are 5in x 10in.