京都府立医科大学 前期 1992年度 問4

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大学名 京都府立医科大学
学科・方式 前期
年度 1992年度
問No 問4
学部 医学部
カテゴリ 微分法の応用 ・ 積分法 ・ いろいろな曲線
状態 解答なし 解説なし ウォッチリスト

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%=========================================================== % ■点A(t, t^3)とか % #1 は点の名前(「A」とか) % #2 は「(t,\,t^3)」とか「(s,\,t,\,u)」と( )内を全部書く % ()の大きさとか,の後のスペースとか微調整ができるように % %----------------------------------------------------------- \providecommand{\点}[2]{${\rm#1} #2$} %=========================================================== %=========================================================== % ■ 証明終了記号 ■ %----------------------------------------------------------- \newcommand{\■}{{\tiny \text{■}}} %=========================================================== \begin{document} %%%%% ■ 本文開始 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{FRAME}% この間に設問を書く% \usepackage{ custom_suseum} をゆうこうにすること \begin{framed} % %---------------------------------------------------------------- % \begin{minipage}[t]{.6\linewidth} \vspace{0mm} $xyz$座標空間において,$yz$平面上の曲線\\ $z=3\left(1-\Dfrac1{y^2}\right)\ (1\LEQQ y \LEQQ3)$ を$z$軸のまわりで回転させて得られる曲面を側面とし, $xy$平面の円盤$x^2+y^2\LEQQ1$を底 とする容器があり,$xy$平面は水平であるとする. いま,この容器に水を時間当り一定の割合で注ぎ込む. 一方,水はその水面の面積に比例して蒸発するものとする. すなわち,時刻$t$での,容器内の水の量を$V(t)$, その水面の面積を$S(t)$, その深さを$h(t)$とするとき,つぎの式が成り立つ. \end{minipage} \begin{minipage}[t]{.35\linewidth} \vspace{0mm} %WinTpicVersion4.28b {\unitlength 0.1in \begin{picture}( 15.7500, 19.4000)( 12.5000,-23.1000) % VECTOR 2 0 3 0 Black White % 2 2025 1820 2025 420 % \special{pn 8}% \special{pa 2026 1820}% \special{pa 2026 420}% \special{fp}% \special{sh 1}% \special{pa 2026 420}% \special{pa 2006 488}% \special{pa 2026 474}% \special{pa 2046 488}% \special{pa 2026 420}% \special{fp}% % VECTOR 2 0 3 0 Black White % 2 2025 1820 1275 2310 % \special{pn 8}% \special{pa 2026 1820}% \special{pa 1276 2310}% \special{fp}% \special{sh 1}% \special{pa 1276 2310}% \special{pa 1342 2290}% \special{pa 1320 2282}% \special{pa 1320 2258}% \special{pa 1276 2310}% \special{fp}% % ELLIPSE 2 0 3 0 Black White % 4 2025 1820 2265 1925 2265 1925 2265 1925 % \special{pn 8}% \special{ar 2026 1820 240 106 0.0000000 6.2831853}% % ELLIPSE 2 0 3 0 Black White % 4 2025 1545 2341 1671 2341 1671 2341 1671 % \special{pn 8}% \special{ar 2026 1546 316 126 0.0000000 6.2831853}% % ELLIPSE 2 0 3 0 Black White % 4 2025 1120 2777 1384 2777 1384 2777 1384 % \special{pn 8}% \special{ar 2026 1120 752 264 0.0000000 6.2831853}% % SPLINE 2 0 3 0 Black White % 4 2260 1842 2340 1562 2477 1328 2477 1328 % \special{pn 8}% \special{pa 2260 1842}% \special{pa 2268 1810}% \special{pa 2282 1748}% \special{pa 2290 1716}% \special{pa 2306 1656}% \special{pa 2316 1624}% \special{pa 2328 1594}% \special{pa 2340 1566}% \special{pa 2352 1536}% \special{pa 2366 1508}% \special{pa 2414 1424}% \special{pa 2432 1398}% \special{pa 2478 1328}% \special{fp}% % SPLINE 2 0 3 0 Black White % 4 1790 1845 1710 1565 1573 1330 1573 1330 % \special{pn 8}% \special{pa 1790 1846}% \special{pa 1784 1814}% \special{pa 1770 1752}% \special{pa 1754 1688}% \special{pa 1724 1598}% \special{pa 1712 1568}% \special{pa 1684 1512}% \special{pa 1670 1482}% \special{pa 1654 1456}% \special{pa 1636 1428}% \special{pa 1602 1374}% \special{pa 1584 1346}% \special{pa 1574 1330}% \special{fp}% % STR 2 0 3 0 Black White % 4 2090 465 2090 515 2 0 0 0 % $z$ \put(20.9000,-5.1500){\makebox(0,0)[lb]{$z$}}% % STR 2 0 3 0 Black White % 4 1250 2168 1250 2218 2 0 0 0 % $x$ \put(12.5000,-22.1800){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 Black White % 4 2610 2120 2610 2170 2 0 0 0 % $y$ \put(26.1000,-21.7000){\makebox(0,0)[lb]{$y$}}% % VECTOR 2 0 3 0 Black White % 2 2025 1820 2825 2060 % \special{pn 8}% \special{pa 2026 1820}% \special{pa 2826 2060}% \special{fp}% \special{sh 1}% \special{pa 2826 2060}% \special{pa 2768 2022}% \special{pa 2774 2046}% \special{pa 2756 2060}% \special{pa 2826 2060}% \special{fp}% % SPLINE 2 0 3 0 Black White % 6 2225 1885 2300 1602 2478 1325 2570 1248 2722 1212 2722 1212 % \special{pn 8}% \special{pa 2226 1886}% \special{pa 2238 1822}% \special{pa 2246 1790}% \special{pa 2252 1760}% \special{pa 2268 1698}% \special{pa 2278 1666}% \special{pa 2298 1606}% \special{pa 2310 1578}% \special{pa 2324 1548}% \special{pa 2338 1520}% \special{pa 2354 1492}% \special{pa 2372 1464}% \special{pa 2388 1438}% \special{pa 2408 1412}% \special{pa 2426 1386}% \special{pa 2448 1360}% \special{pa 2468 1336}% \special{pa 2514 1290}% \special{pa 2540 1268}% \special{pa 2566 1252}% \special{pa 2594 1238}% \special{pa 2624 1228}% \special{pa 2688 1216}% \special{pa 2720 1212}% \special{pa 2722 1212}% \special{fp}% \end{picture}}% \end{minipage} \[\Dfrac{dV(t)}{dt}=A-kS(t) \quad (ただし,Aとkは正の定数である.)\] このとき, \begin{enumerate} \item $h(t)$の満す微分方程式を求めよ. \item $A=3\pi$,$k=1$のとき,$h(0)=0$を満す$h(t)$を求めよ. \item (2)の$h(t)$について,$\dlim_{t\to\infty}h(t)$を求めよ. \end{enumerate} %-------------------------------------------------------------- \end{framed} %\end{FRAME} %--- 解答 ------------------------------------------------------------------------ {\footnotesize }% footnotesize %%%%%%% ■ 本文終了 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}