AMC 8 2023 Problem 12
AMC8 2023 年第 12 题
圆★★★☆☆
The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?
如图所示，在一个大的白色圆圈内部有许多较小的白色和灰色圆圈．问灰色部分在大的白色圆圈内部所占的比例是多少？
- A.
- B.
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- D.
- E.
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
First, the total area of the radius 3 circle is simply just when using our area of a circle formula. Now from here, we have to find our shaded area. This can be done by adding the areas of the -radius circles and add; then, take the area of the 1 radius circles and subtract that from the area of the 2 radius circle to get our resulting complex shape area. Adding these up, we will get . So, our answer is . Minor edits by Minor edits by Chowdhury
Pretend each circle is a square. The large shaded circle is a square with area $16
~\text{units}^2$, and the two white circles inside it each have areas of $4~\text{units}^2$, which adds up to $8~\text{units}^2$. The three small shaded circles become three squares with area $1~\text{units}^2$, and add up to $3~\text{units}^2$. Adding the areas of the shaded circles (19) and subtracting the areas of the white circles (8), we get $11~\text{units}^2$. Since the largest white circle in which all these other circles are becomes a square that has area $36~\text{units}^2$, our answer is $\dfrac{11}{36}$. -claregu LaTeX (edits -apex304, CoOlPoTaToEs, KGINSPECTORBOI, SlimeKnight, rickastleylover99)
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