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[킹앤카&킹앤포&킹앤유] 퀴5. 달리기 경주

수학동아 2017.05.24 02:19 조회 1027

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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

KASIT 수학문제연구회 학생들이 출제한 수리 논술 문제입니다. 문제를 잘 풀어준 학생을 뽑아 수학문제연구회 학생들이 직접 답안을 첨삭해줄 예정입니다. 문제를 풀어 보세요!

5월에는 봄 운동회가 열린다. 그런데 운동회의 꽃인 달리기 경기를 하기에는 트랙이 너무 짧다. 달리기 경기를 하려면 전체 트랙 둘레의 길이가 300m는 돼야 한다. 학교는 트랙을 바깥 방향으로 얼마나 늘려야 할지 수학문제연구회에 의뢰했다. 트랙 양 끝의 모양이 반원이라 할 때, 어떻게 해야 할까? 또, 늘어난 트랙 위에서 6m/s로 달리는 성연이와 5m/s로 달리는 범석이가 6000m 달리기 시합을 한다고 할 때, 성연이는 범석이를 몇 번 따라잡을까? 단, 성연이와 범석이는 같은 위치에서 출발한다.

다음글 퀴6. 카리용의 높이 구하기

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디듀우 Lv.7 2017.07.14 18:35

가운데 직사각형 부분의 가로를 7.2m 늘리면 트랙이 300m가 됩니다. 원주율을 3.14로 놓으면요.

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디듀우 Lv.7 2017.07.14 18:37

성연이가 초당 1m씩 앞서가니까 다시 따라잡는데는 300초가 걸리고, 성연이가 1000초 동안 트랙 위를 달리니 3번까지 따라잡을 수 있겠네요.

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제네릭 Lv.1 2017.07.17 17:01

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꼭 직사각형 부분만의 길이를 변경할 필요는 없습니다. 모든 방향으로 트랙을 크게 한다고 생각하고,

반원의 반지름이 커지는 정도를 $r (r > 0)$ , 직사각형의 가로 길이가 커지는 정도를 $l (l > 0)$ 이라고 둡시다.

수식을 유도하면,
$2\pi(20+r) + 2(80 + l) = 300$ ,

$= 40\pi + 2\pi r + 160 + 2l = 300$

$= 40\pi + 2\pi r + 2l = 140$

= $= 2\pi r + 2l = 140 - 40\pi$

$\therefore l = -\pi (r + 20) + 70$

$l$ 에 대해 그래프를 나타내도록 합시다. 앞서 제시한 조건( $l, r$ 은 0보다 큰 양수)를 만족해야 함에 유의합시다.

Wolfram Alpha를 이용해 그래프를 그려보니, x절편은 2.28에 근사하며, y절편은 7에 근사합니다.

즉, 이 그래프에서 x > 0, y > 0인 영역 내에서 둘레의 길이가 300m가 되도록 할 수 있습니다.

2번 문제는 디듀우님이 잘 풀어주셔서 별달리 할 말이 없네요 ^_^;;

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