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数学 高校生

このマーカーで引いたところって、Q中心の半径1の円で-2t分回転させたからこのような座標隣っているのですか?

重要 例題 287 曲線の長さ (2) 円C:x+y2=9の内側を半径1の円Dが滑らずに転がる。時刻 t において D は点 (3cost, 3sint) でCに接している。 (1) 時刻 t=0 において点 (3,0)にあったD上の点Pの時刻t における座標 (x(t),y(t)) を求めよ。ただし, 0≦t≦πとする。 2 X(2) (1) の範囲で点Pの描く曲線の長さを求めよ。 [類 早稲田大〕 基本286 指針 (1) ベクトルを利用。 PはDの円周上にあり, Dの中心Qとともに動く。 そこで OP=OQ+QP (Oは原点)として, QP をもの式で表す。 Q, 毎日 円x2+y2=2(x>0)の周上の点Pの座標は (rcost, rsint) で表され,このとき OP がx軸の正の方向となす角はtである。 dx (2) p.465 基本事項 ① S. √ (d) + (a)* dy Ja V dt dt 解答 (1) A(3, 0), T(3 cost, 3sint) 3. 00107: DとCがTで接しているとき, Dの中心Qの座標は (2cost, 2sint) である。 また, TP=TA=3tである から,x軸の正の方向から半直線 QP への角は t-3t=-2t よって 0を原点とすると OP=OQ+QP introst ( = 16 sin²³-t 2 dt の公式を利用。 (2cost2sint)+(cos(2t), sin(-2t))ヶ =(2cost+cos2t, 2sint-sin2t) (2) x(t)=-2sint-2sin2t, y' (t)=2cost-2cos2t から {x' (t)}²+{y'(t)}²=4(sin²t+2 sintsin2t+sin²2t)=1 +4(cos²t-2 cost cos 2t+ cos²2t) =4(2-2cos3t)=8 (1-cos3t) よって、求める曲線の長さは 3 3 St / 16sin222tdt = S." asin 2/2 tdt 10 大 0905 YA 3 C D St 3 = =4・ -4. [-cos/211³-¹6) ・COS ・土 3 2 0 $3+$1 Q 3t 0≤t≤ 2012/2πであるから sin ²01² 3 T(3cost, 3sint) (0²2) 5 (1) ²2=(²²+²²= < sin20+ cos20=1, costcos 2t-sintsin2t =cos(t+2t) 半角の公式により -2t3 AX T 2004: 点Pの描く曲線はハイポ サイクロイドである(p.137 でα=3、b=1の場合)。 1-cos 3t =sin²t 2 RCK TO 100 4467 ◄S³* √ {x' (t)}²³+ {y'(t)}² dt 練習 a>0とする。 長さ2maのひもが一方の端を半径aの円周上の点Aに固定して, ©287 その円に巻きつけてある。このひもを引っ張りながら円からはずしていくとき, ひもの他方の端 P が描く曲線の長さを求めよ。 8章 41 曲線の長さ、速度と道のり 下移動

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英語 高校生

これの100字要約日本語でしていただけませんか?

5 19 A concerted drive to reduce obesity in one Australian town resulted in a whole generation of slimmer, faster, and healthier children, researchers reported yesterday. They said that the program, a simple mixture of persuasion and (A)incentives, was astonishingly successful. It led to 2,000 children gaining less weight, watching far less television, taze (and playing more sports. The "Be Active, Eat Well" project, conducted by Deakin University in the small town of Colac, 150 km southwest of Melbourne, ended with Colac's children weighing an average of one kilogram less than the norm for Australian children of their age. Their waistlines were an ウェスト average of cm smaller - 2 cm for boys and 4 cm for girls. Professor Boyd Swinburn from Deakin University in Melbourne said yesterday that the Colac experiment had proved to be "astonishingly successful." It was the first such program in the world to report significant reductions in waistline and weight. Professor Swinburn said: "Most people would think individual weight loss of one kilogram is not much, but here we're talking about shifting the weight of a couple of thousand kids, and 15 that's actually quite (B) phenomenal. In fact, across a population, that is absolutely huge." The experiment began three years ago when the university researchers descended on Colac's population of about 10,000 people, urging parents, teachers, doctors, and local fast-food outlets to support changes for all children aged between 4 and 12. The program included opening up more after-school activity centers for children and introducing 20 brightly colored lunch packs that contained a pitta salad wrap*¹ and fruit tub2. Parents were encouraged to (c) monitor strictly the amount of time their children watched television or walk or cycle to They were asked to encourage their children spent on computers. (3) school rather than drive them. While the researchers had hoped to cut television viewing by 10 percent, the final results 25 reported children's television viewing had dropped by 21 percent and soft drink consumption by 70 percent. There was an increase of almost 70 percent in the number of children participating in after-school sports. 10 7. ★★★ 参照チェックノート p.38 414 words 56 早稲田大学 Even the town's fish and chip shop owner switched from using animal fats to sunflower oil. He reduced the saturated fats3 in chips from 49 percent to 9.1 percent. The other fast-food outlets 30 also switched from animal fats, leading to a cut in saturated fats consumed in the town of 55 kg a week. Adults then began to follow their children's example, and the local self-defense academy went from 16 members to 75. pitta satu 1 (A (

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