AMC 8 2018 Problem 15
AMC8 2018 年第 15 题
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In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of 1 square unit, then what is the area of the shaded region, in square units?
如下图所示，2 个小圆各自的直径都是大圆的半径，如果这 2 个小圆的面积之和是 1 平方单位，那么图中阴影部分的面积是多少平方单位？
- A.
- B.
- C.
- D.1
- E.
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Let the radius of the large circle be . Then, the radius of the smaller circles are . The areas of the circles are directly proportional to the square of the radii, so the ratio of the area of the small circle to the large one is ( is of .) This means the combined area of the 2 smaller circles is half of the larger circle, and therefore the shaded region is equal to the combined area of the 2 smaller circles, which is 1.
Let the radius of the two smaller circles be . It follows that the area of one of the smaller circles is . Thus, the area of the two inner circles combined would evaluate to which is 1. Since the radius of the bigger circle is two times that of the smaller circles (the diameter), the radius of the larger circle in terms of would be . The area of the larger circle would come to . Subtracting the area of the smaller circles from that of the larger circle (since that would be the shaded region), we have Therefore, the area of the shaded region is 1.
The area of a small circle is . Solving, we get . The radius of the large circle is R=2r. The area of the large circle is . Subtract the area of the small circles from the area of the large circle to get the area of the shaded region: 2-1=1. To get the area of the small circles, we can get the equation . Solving for , we get . Then, we can get the radius of the big circle by doubling the small circle's radius, and that gives . Because the area of a circle is , you'll have to square the answer and multiple it by . After you square it, you'll get , which equals to . Since the area of a circle is , we get , and that equals , which equals 2. Since each of the circles' area is , the combined area of the small circles is 1. Since 2-1=1, the area of the shaded region is 1.
Since the smaller circles are equivalent, the area of each of them is square unit. Since the diameter of each of the two smaller circles is a radius of the larger circle, their areas must be in a ratio of . This tells us that the area of the larger circle is double the area of the smaller circle, giving us .
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