Continued Square Roots

Continued Square Roots は直訳すると連続平方根となりますが，連続根号数とも訳されています．Wolfram MathWorldで探すと，Nested Radical（多重根号） にLinkされ，次の式が書かれていますが，これは正確にいうとInfinite Nested Radical（無限多重根号）ですね．$$\displaystyle \lim_{ k \to \infty }x_0+\sqrt{x_1+\sqrt{x_2+\sqrt{...+x_k}}}$$もちろん根号の数が有限な例としては$\sqrt{2+\sqrt{3}}$などがあります．上の式で$x_0=0$とし，それ以降の$x_k=n$とすると次式になります．$$\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+...}}}}$$これが無限に続くとどんな値に近づくのでしょう．$$\sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+...}}}}=x$$とおくと，いちばん外側の$\sqrt{\quad}$の中の$n+$以降は$x$と等しいので，$$\sqrt{n+x}=x$$と考えられ，両辺を平方して移項すると，$$x^2-x-n=0$$この解は正の数なので，$$x=\frac{1+\sqrt{1+4n}}{2}$$例えば$n=1$なら，$$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}=x$$とおくと，$$\sqrt{1+x}=x$$となり，両辺を平方して移項すると，$$x^2-x-1=0$$これの正の解は$x=\frac{1+\sqrt{5}}{2}$，すなわち黄金比の値になります．

Mathematics for the international student  Pre-Diploma SL and HL (MYP 5 Plus) Presumed Knowledge for SL and HL courses (HAESE Mathematics)

またこの図のように$n=2$なら，$$\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}=x$$とおくと，$$\sqrt{2+x}=x$$となり，両辺を平方して移項すると，$$x^2-x-2=0$$これの正の解は$x=2$となります．上図の INVESTIGATION 2 はこの値に近づくことを調べさせています．

[Continued Square Roots 問題]
Find the value of x.$$\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+...}}}}=x$$解答はこちら

[Reference]

Mathematics for the international student
Pre-Diploma SL and HL (MYP 5 Plus) Presumed Knowledge for SL and HL courses
(HAESE Mathematics)

Continued Square Root
http://mathworld.wolfram.com/ContinuedSquareRoot.html

Ferris Wheel

Ferris Wheel

Ferris wheelは数学用語ではありませんが，三角関数の応用問題で頻出するので，ぜひ知っておきたい用語です．

Ferrisには特に意味はなく，人名のひとつなので，Ferris wheelというと｢フェリスさんの車輪｣という意味になりそうですが，これはFerrisという人が世界初の観覧車(Chicago Wheel)を設計したので，それに敬意を表して観覧車のことを一般的にこう呼んでいます．

英国系では観覧車をbig wheel，または巨大なものをgiant wheelと呼ぶこともありますが，英国の出版社であるCambridge University Press，Oxford University Press，Pearson EducationのIB（国際バカロレア）数学の教科書など，多くの場合はFerris wheelという用語が使われています．

Cambridge University Press
The original Ferris Wheel was constructed in 1893 in Chicago. It was just over 80 m tall and could complete one full revolution in 9 minutes. During each revolution, how much time did the passengers spend more than 50 m above the ground? (IB Math SL)

Oxford University Press
A Ferris wheel at an amusement park reaches a maximum height of 46 metres and a minimum height of 1 metre. It takes 20 minutes for the wheel to make one full rotation. Write a sine function to model the height of the child t minutes after boarding the Ferris wheel. (IB Math SL)

Pearson Education
We are surrounded by periodic functions. A few examples include: the average daily temperature variation during the year; sunrise and the day of the year; animal populations over many years; the height of tides and the position of the Moon; and your height above ground when riding a Ferris wheel and the rotation of the wheel. (IB Math SL)

[Ferris Wheel 問題]
The Ferris wheel at the Royal Show turns one full circle every minute. The lowest point is 1 metre from the ground, and the highest point is 25 metres above the ground.
(a) When riding on the Ferris wheel, your height above ground level after t seconds is given by the model h(t)=a+bsin(c(t-d)). Find the values of a, b, c, and d given that you start your ride at the lowest point.
(b) If the motor driving the Ferris wheel breaks down after 91 seconds, how high up would you be while waiting to be rescued?
(IB Math SL : Haese Mathematics)

解答はこちら

[Reference]
IB Math text books by Oxford, Cambridge, Pearson and Haese