AMC 8 2004 Problem 23
AMC8 2004 年第 23 题
行程问题★★★☆☆
Tess runs counterclockwise around rectangular block . She lives at corner . Which graph could represent her straight-line distance from home?
Tess 沿着长方形方块 JKLM 的边逆时针跑步。她住在角落 J，那么下面哪张图可以表示她距离家的直线距离？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
For her distance to be represented as a constant horizontal line, Tess would have to be running in a circular shape with her home as the center. Since she is running around a rectangle, this is not possible, ruling out and with straight lines. Because is the diagonal of the rectangle, and is at the middle distance around the perimeter, her maximum distance should be in the middle of her journey. The maximum in is at the end, and has two maximums, ruling both out. Thus the answer is.
Even though may find an answer, it is never proven that actually describes the distance to Tess' home.
It may be easier to compare the distance traveled by Tess rather than the time spent, if we suppose that Tess moves at constant speed then the graphs distance-time and distance-position are equivalent.
Let be the linear distance walked by Tess, and her distance from home . We will divide this problem into four parts (sections , , and ).
Section : At any point of the segment , the distance of Tess from home is described by the function where is the linear distance traveled by Tess from point .
Section : From to , the distance from home is described by a right triangle, where is the hypotenuse. In this situation the distance follows the function , which is a curve (this was obtained by applying the Pythagorean Theorem).
Section : Following an analogue reasoning to Section , the distance function in this case results which is also a curve.
Section : Analogue to Section , we obtain a linear function
None of the shown answers in the multiple choice describe the real distance between Tess and home along the path.
The real graph should have a shape of the following kind:
Which is slightly different from the expected solution .
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