1983 AMC 12 第 27 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
27.
在一个晴天，一个大球体放在水平地面上。某一时刻，球体的影子从球体与地面的接触点起延伸 米。同一时刻，一根米尺竖直放置，一端接触地面，投下长度为 米的影子。该球体的半径是多少米？（假设太阳光线互相平行，并将米尺视为线段。）
A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length m. What is the radius of the sphere in meters? (Assume the sun’s rays are parallel and the meter stick is a line segment.)
小提示：
在竖直截面中，影子边界处的太阳光线与一个圆相切
In a vertical cross-section, the limiting sun ray is tangent to a circle
大提示：
米尺表明光线每水平前进 个单位就竖直上升 个单位
The meter stick shows that the ray rises unit for every horizontal units
解答：
以球体与地面的接触点为 ，圆心为 。影子边界处的太阳光线经过影子端点 ，斜率为 ，所以方程为 。相切意味着点 到这条直线的距离等于 ：因此 。
所以正确答案是 E。
Take the sphere’s ground-contact point as and its center as The limiting sun ray passes through the shadow endpoint and has slope so its equation is Tangency means the distance from to this line equals Hence
Therefore, the correct answer is E.
其他年份的第 27 题
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- 1995 AMC 12
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- 1999 AMC 12