Temperature Change in an Insulated House
- Find the times when the temperature in an insulated house drops by 2.1°C based on the function T = 3cos (πt/12)
- The temperature change in the house during a 24-hour period can be modeled by the function T = 3cos (πt/12), where t is the number of hours since midday.
- The temperature drops by 2.1°C between 9pm and 3am based on the given function, which represents a 24-hour period with 24 cycles.
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T = 3cos (πt/12) 問題英文です
ちょっと混乱しています。 ハッキリしたいのでこちらで質問させて下さい。 問題 An insulated house maintains an average temperature in summer of 20°．The change in temperature T° C in the house during a 24-hour period can be modelled by the function. T = 3cos (πt/12) where t is the number of hours since midday. At what times does the temperature in the house drop by 2.1° (T=-2.1°)？The angle is in radians. 添付したグラフを見ると答えは９ｐｍと３amの間、というのはわかります。 混乱しているのは例えば ｙ = 3cos (３ｘ)だったら周期は２π/3で ０から２πの間に３サイクルあると理解しています。 この場合、T = 3cos (πt/12) なので２π/(π/12) = 24 となりサイクルが２４ある、と考えるのです。 で、この場合２πではなく２４－hour period となっているので２πの中にサイクルが２４回ではなく２４－hourの中にサイクルが２４回あると考えます。なので１時間毎に１サイクルあると考えてしまうのです。 しかしこのグラフを見ると２４－hour period にサイクルは一度しかありません。 質問したい事が分かって頂ければ幸いです。。。 私のどこが間違った考え方なのか指摘して頂けませんか？ 宜しくお願い致します。
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要はサイクルfと角周波数ωと周期Tをしっかり区別することです。これらの間には ω=２πf (1) fT=1 (2) の関係があります。 時間をtと記すと三角関数を用いた周期運動は y=asin[ω(t-t0)]=asin[2πf(t-t0)] (3) または y=acos[ω(t-t0)]=acos[2πf(t-t0)] (4) で表されます。 a:振幅、t0：時間遅れ です。 時間単位としてはふつう、秒をもちいますが、この問題では時間(hour)を用いることにします。 周期の単位は時間、サイクルの単位は１/時間、つまり１時間の中で何回ということです。 ＞混乱しているのは例えば ｙ = 3cos (３ｘ)だったら周期は２π/3で ０から２πの間に３サイクルあると理解しています。 記号を整理するためｘの代わりにtを用います。 ｙ = 3cos (３ｘ) (4)と比較して a=3,t0=0,ω=3 a,t0は検討対象外として ω=２πf＝3 f=3/2π T=1/f=2π/3 よって質問者の「周期は２π/3で」というのはあっています。「０から２πの間に３サイクルある」という気持ちはわかりますが正しくは「角周波数は3、またはサイクルは3/2π」というべきです。 ＞この場合、T = 3cos (πt/12) なので２π/(π/12) = 24 となりサイクルが２４ある、と考えるのです。 周期のTと混同しないためTemperature(温度)=Tempを用いて Temp= 3cos (πt/12) と記すことにします。 (4)と比較して ω=π/12 f=ω/2π＝1/24 T=1/ｆ＝24 です。よって周期が24時間、サイクルは1/24（/時間)すなわち２４時間に一回 が正しい認識です。
その他の回答 (1)
- trytobe
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T = 3cos (πt/12) で ラジアン表記の部分が、 t=0 のときに 0π/12 ＝ 0、t=24hr のときに 24π/12 ＝ 2π/1 ですから、 24hr で 1 サイクル、という考えで、合っているのです。
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