For this question, please reference the image at:
imgur<dot>com/KSYAN6T
All circles in the dartboard have the same center.
Each circle has half the radius of the previous circle.
The colors alternate orange-white-orange-white-etc. forever.
How much of the dartboard is colored orange?
the radii form a geometric sequence
r, r/2, r/4, ...
The orange rings have areas that form another sequence:
πr^2 - π(r/2)^2, π(r/4)^2 - π(r/8)^2, ...
= πr^2 (3/4 + 3/64 + ...)
the sum of all those areas is thus
πr^2 (3/4)/(1 - 1/16) = 4/5 πr^2
so, 4/5 of the dartboard is orange
To determine how much of the dartboard is colored orange, we need to understand the relationship between the radii of the circles and their corresponding areas. From the given information, we know that each circle has half the radius of the previous circle.
To calculate the area of each circle, we can use the formula A = πr², where A represents the area and r represents the radius.
Let's consider the first circle, which is fully colored orange. We'll call its radius r. The area of this circle is A₁ = πr².
The second circle, which is white, has half the radius of the first circle. Therefore, its radius is (1/2)r. The area of this circle is A₂ = π(1/2r)² = π(1/4)r².
The third circle, which is orange, has half the radius of the second circle. Hence, its radius is (1/4)r. The area of this circle is A₃ = π(1/4r)² = π(1/16)r².
The pattern continues, with each subsequent circle having a radius that is half of the previous circle. Therefore, the area of each subsequent orange circle can be expressed as a fraction of the initial circle's area.
To calculate the total orange area, we need to sum the areas of all the orange circles. Since the pattern continues forever, we can express the total orange area as an infinite geometric series.
The sum of an infinite geometric series can be calculated using the formula S = a / (1 - r), where S represents the sum, a represents the first term, and r represents the common ratio. In this case, the first term, a, is the area of the first orange circle, and the common ratio, r, is (1/4)² = 1/16.
Substituting the values into the formula, we have S = (πr²) / (1 - 1/16).
Now, we have all the information we need to calculate the total orange area. However, we need the value of r, which is not given in the text. To obtain the value of r, please provide the diameter or the length of the side of the largest circle in the dartboard, if available.