大阪大学 前期理系 1992年度 問4

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解答作成者: 森 宏征

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入試情報

大学名 大阪大学
学科・方式 前期理系
年度 1992年度
問No 問4
学部 理学部 ・ 医学部 ・ 歯学部 ・ 薬学部 ・ 工学部 ・ 基礎工学部
カテゴリ 微分法の応用 ・ 積分法の応用
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn]{jreport} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{graphicx} \usepackage{delarray} \usepackage{multicol} \usepackage{amscd} \usepackage{pifont} \usepackage{color} \ExecuteOptions{usename} \usepackage{vector3} %\usepackage{myhyper} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} 図のような容器を考える. 空の状態から始めて, 単位時間一定の割合で水を注入し, 底から測った水面の高さ $h$ が10になるまで続ける. 水面の上昇する速さ $v$ は, 水面の高さ $h$ の関数として, \[ v = \frac{\sqrt{\vphantom{b} 2 + h}}{\log(2 + h)} \quad (0 \leqq h \leqq 10) \] で与えられるものとする. 水面の上昇が始まってから水面の面積が最大となるまでの時間を求めよ. \hfill(配点率20%) \begin{center} %\input{osaka92s4f_zu_2} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 12.6900, 17.5400)( 13.2900,-22.2500) % ELLIPSE 1 0 3 0 % 4 1624 528 1849 585 1849 585 1849 585 % \special{pn 13}% \special{ar 1624 528 226 58 0.0000000 6.2831853}% % ELLIPSE 1 0 3 0 % 4 1846 1668 2227 1743 2227 1743 2227 1743 % \special{pn 13}% \special{ar 1846 1668 382 76 0.0000000 6.2831853}% % SPLINE 1 0 3 0 % 29 1849 533 1828 558 1817 565 1802 573 1786 581 1780 592 1772 614 1786 636 1828 651 1860 655 1927 673 1964 691 2004 731 2041 794 2071 864 2089 912 2111 994 2130 1063 2141 1112 2155 1177 2180 1311 2199 1411 2217 1543 2225 1624 2233 1724 2236 1807 2236 1871 2236 1996 2225 2176 % \special{pn 13}% \special{pa 1850 534}% \special{pa 1830 558}% \special{pa 1802 574}% \special{pa 1780 596}% \special{pa 1776 626}% \special{pa 1802 646}% \special{pa 1834 652}% \special{pa 1864 656}% \special{pa 1896 664}% \special{pa 1926 674}% \special{pa 1956 686}% \special{pa 1982 706}% \special{pa 2002 730}% \special{pa 2020 756}% \special{pa 2036 784}% \special{pa 2050 814}% \special{pa 2062 842}% \special{pa 2074 872}% \special{pa 2086 902}% \special{pa 2096 932}% \special{pa 2104 964}% \special{pa 2112 994}% \special{pa 2120 1026}% \special{pa 2128 1056}% \special{pa 2136 1088}% \special{pa 2142 1118}% \special{pa 2150 1150}% \special{pa 2156 1182}% \special{pa 2162 1212}% \special{pa 2168 1244}% \special{pa 2174 1276}% \special{pa 2180 1308}% \special{pa 2186 1338}% \special{pa 2192 1370}% \special{pa 2198 1402}% \special{pa 2204 1432}% \special{pa 2208 1464}% \special{pa 2212 1496}% \special{pa 2216 1528}% \special{pa 2220 1560}% \special{pa 2222 1592}% \special{pa 2226 1624}% \special{pa 2228 1656}% \special{pa 2230 1688}% \special{pa 2234 1720}% \special{pa 2234 1752}% \special{pa 2236 1784}% \special{pa 2236 1816}% \special{pa 2236 1848}% \special{pa 2236 1880}% \special{pa 2236 1912}% \special{pa 2236 1944}% \special{pa 2236 1976}% \special{pa 2236 2008}% \special{pa 2236 2040}% \special{pa 2234 2072}% \special{pa 2232 2104}% \special{pa 2228 2134}% \special{pa 2226 2166}% \special{pa 2226 2176}% \special{sp}% % SPLINE 1 0 3 0 % 32 1414 555 1443 573 1454 602 1459 621 1454 644 1448 655 1422 673 1399 684 1381 699 1360 725 1344 754 1334 787 1330 828 1330 864 1330 898 1330 957 1337 1019 1344 1086 1356 1167 1367 1259 1374 1303 1385 1370 1399 1421 1414 1494 1429 1539 1443 1598 1459 1664 1480 1750 1510 1845 1543 1941 1577 2056 1617 2169 % \special{pn 13}% \special{pa 1414 556}% \special{pa 1442 572}% \special{pa 1454 602}% \special{pa 1458 632}% \special{pa 1444 660}% \special{pa 1416 676}% \special{pa 1388 692}% \special{pa 1366 716}% \special{pa 1350 744}% \special{pa 1338 774}% \special{pa 1332 804}% \special{pa 1330 836}% \special{pa 1330 868}% \special{pa 1330 900}% \special{pa 1330 932}% \special{pa 1332 964}% \special{pa 1334 996}% \special{pa 1338 1028}% \special{pa 1342 1060}% \special{pa 1346 1092}% \special{pa 1350 1124}% \special{pa 1354 1156}% \special{pa 1358 1188}% \special{pa 1362 1218}% \special{pa 1366 1250}% \special{pa 1372 1282}% \special{pa 1376 1314}% \special{pa 1380 1346}% \special{pa 1388 1376}% \special{pa 1396 1408}% \special{pa 1404 1438}% \special{pa 1408 1470}% \special{pa 1416 1502}% \special{pa 1426 1532}% \special{pa 1436 1562}% \special{pa 1442 1594}% \special{pa 1450 1624}% \special{pa 1458 1656}% \special{pa 1464 1688}% \special{pa 1472 1718}% \special{pa 1480 1750}% \special{pa 1488 1780}% \special{pa 1498 1810}% \special{pa 1508 1840}% \special{pa 1520 1872}% \special{pa 1530 1902}% \special{pa 1540 1932}% \special{pa 1550 1962}% \special{pa 1558 1992}% \special{pa 1568 2024}% \special{pa 1576 2054}% \special{pa 1586 2084}% \special{pa 1598 2114}% \special{pa 1608 2144}% \special{pa 1618 2170}% \special{sp}% % LINE 3 0 3 0 % 60 1971 1598 1828 1742 1949 1598 1805 1742 1927 1598 1783 1742 1909 1595 1765 1739 1886 1595 1742 1739 1864 1595 1720 1739 1842 1595 1702 1735 1820 1595 1680 1735 1798 1595 1662 1731 1775 1595 1639 1731 1749 1598 1621 1727 1728 1598 1602 1724 1705 1598 1584 1720 1680 1602 1565 1716 1654 1606 1547 1713 1628 1609 1529 1708 1602 1613 1514 1701 1577 1617 1496 1698 1547 1624 1480 1690 1518 1631 1469 1679 1990 1602 1849 1742 2011 1602 1872 1742 2030 1606 1893 1742 2052 1606 1915 1742 2071 1609 1941 1739 2089 1613 1964 1739 2108 1617 1990 1735 2126 1620 2011 1735 2144 1624 2037 1731 2162 1627 2063 1727 % \special{pn 4}% \special{pa 1972 1598}% \special{pa 1828 1742}% \special{fp}% \special{pa 1950 1598}% \special{pa 1806 1742}% \special{fp}% \special{pa 1928 1598}% \special{pa 1784 1742}% \special{fp}% \special{pa 1910 1596}% \special{pa 1766 1740}% \special{fp}% \special{pa 1886 1596}% \special{pa 1742 1740}% \special{fp}% \special{pa 1864 1596}% \special{pa 1720 1740}% \special{fp}% \special{pa 1842 1596}% \special{pa 1702 1736}% \special{fp}% \special{pa 1820 1596}% \special{pa 1680 1736}% \special{fp}% \special{pa 1798 1596}% \special{pa 1662 1732}% \special{fp}% \special{pa 1776 1596}% \special{pa 1640 1732}% \special{fp}% \special{pa 1750 1598}% \special{pa 1622 1728}% \special{fp}% \special{pa 1728 1598}% \special{pa 1602 1724}% \special{fp}% \special{pa 1706 1598}% \special{pa 1584 1720}% \special{fp}% \special{pa 1680 1602}% \special{pa 1566 1716}% \special{fp}% \special{pa 1654 1606}% \special{pa 1548 1714}% \special{fp}% \special{pa 1628 1610}% \special{pa 1530 1708}% \special{fp}% \special{pa 1602 1614}% \special{pa 1514 1702}% \special{fp}% \special{pa 1578 1618}% \special{pa 1496 1698}% \special{fp}% \special{pa 1548 1624}% \special{pa 1480 1690}% \special{fp}% \special{pa 1518 1632}% \special{pa 1470 1680}% \special{fp}% \special{pa 1990 1602}% \special{pa 1850 1742}% \special{fp}% \special{pa 2012 1602}% \special{pa 1872 1742}% \special{fp}% \special{pa 2030 1606}% \special{pa 1894 1742}% \special{fp}% \special{pa 2052 1606}% \special{pa 1916 1742}% \special{fp}% \special{pa 2072 1610}% \special{pa 1942 1740}% \special{fp}% \special{pa 2090 1614}% \special{pa 1964 1740}% \special{fp}% \special{pa 2108 1618}% \special{pa 1990 1736}% \special{fp}% \special{pa 2126 1620}% \special{pa 2012 1736}% \special{fp}% \special{pa 2144 1624}% \special{pa 2038 1732}% \special{fp}% \special{pa 2162 1628}% \special{pa 2064 1728}% \special{fp}% % LINE 3 0 3 1 % 8 2177 1635 2089 1724 2196 1638 2115 1720 2210 1646 2144 1713 2222 1657 2173 1705 % \special{pn 4}% \special{pa 2178 1636}% \special{pa 2090 1724}% \special{fp}% \special{pa 2196 1638}% \special{pa 2116 1720}% \special{fp}% \special{pa 2210 1646}% \special{pa 2144 1714}% \special{fp}% \special{pa 2222 1658}% \special{pa 2174 1706}% \special{fp}% % ELLIPSE 2 0 3 0 % 4 1919 2169 2229 2225 1504 2169 2296 2169 % \special{pn 8}% \special{ar 1920 2170 310 56 6.2831853 6.2831853}% \special{ar 1920 2170 310 56 0.0000000 3.1415927}% % LINE 3 0 3 0 % 60 2170 2203 1705 1739 2152 2206 1680 1735 2134 2211 1654 1731 2115 2214 1632 1731 2093 2214 1602 1724 2074 2218 1577 1720 2052 2218 1550 1716 2033 2221 1522 1708 2011 2221 1480 1690 1993 2225 1469 1701 1971 2225 1473 1727 1949 2225 1485 1761 1927 2225 1492 1789 1904 2225 1504 1823 1883 2225 1514 1856 1860 2225 1529 1893 1835 2221 1540 1926 1812 2221 1547 1955 1786 2218 1558 1989 1765 2218 1565 2018 1739 2214 1577 2052 1712 2211 1584 2081 1687 2206 1598 2118 1658 2200 1613 2155 2185 2195 1728 1739 2203 2192 1749 1739 2217 2185 1775 1742 2225 2169 1798 1742 2229 2150 1820 1742 2229 2129 1842 1742 % \special{pn 4}% \special{pa 2170 2204}% \special{pa 1706 1740}% \special{fp}% \special{pa 2152 2206}% \special{pa 1680 1736}% \special{fp}% \special{pa 2134 2212}% \special{pa 1654 1732}% \special{fp}% \special{pa 2116 2214}% \special{pa 1632 1732}% \special{fp}% \special{pa 2094 2214}% \special{pa 1602 1724}% \special{fp}% \special{pa 2074 2218}% \special{pa 1578 1720}% \special{fp}% \special{pa 2052 2218}% \special{pa 1550 1716}% \special{fp}% \special{pa 2034 2222}% \special{pa 1522 1708}% \special{fp}% \special{pa 2012 2222}% \special{pa 1480 1690}% \special{fp}% \special{pa 1994 2226}% \special{pa 1470 1702}% \special{fp}% \special{pa 1972 2226}% \special{pa 1474 1728}% \special{fp}% \special{pa 1950 2226}% \special{pa 1486 1762}% \special{fp}% \special{pa 1928 2226}% \special{pa 1492 1790}% \special{fp}% \special{pa 1904 2226}% \special{pa 1504 1824}% \special{fp}% \special{pa 1884 2226}% \special{pa 1514 1856}% \special{fp}% \special{pa 1860 2226}% \special{pa 1530 1894}% \special{fp}% \special{pa 1836 2222}% \special{pa 1540 1926}% \special{fp}% \special{pa 1812 2222}% \special{pa 1548 1956}% \special{fp}% \special{pa 1786 2218}% \special{pa 1558 1990}% \special{fp}% \special{pa 1766 2218}% \special{pa 1566 2018}% \special{fp}% \special{pa 1740 2214}% \special{pa 1578 2052}% \special{fp}% \special{pa 1712 2212}% \special{pa 1584 2082}% \special{fp}% \special{pa 1688 2206}% \special{pa 1598 2118}% \special{fp}% \special{pa 1658 2200}% \special{pa 1614 2156}% \special{fp}% \special{pa 2186 2196}% \special{pa 1728 1740}% \special{fp}% \special{pa 2204 2192}% \special{pa 1750 1740}% \special{fp}% \special{pa 2218 2186}% \special{pa 1776 1742}% \special{fp}% \special{pa 2226 2170}% \special{pa 1798 1742}% \special{fp}% \special{pa 2230 2150}% \special{pa 1820 1742}% \special{fp}% \special{pa 2230 2130}% \special{pa 1842 1742}% \special{fp}% % LINE 3 0 3 1 % 40 2229 2107 1864 1742 2233 2088 1886 1742 2233 2067 1909 1742 2233 2044 1927 1739 2236 2026 1949 1739 2236 2004 1971 1739 2236 1981 1990 1735 2236 1960 2011 1735 2236 1937 2030 1731 2236 1915 2052 1731 2236 1893 2071 1727 2236 1871 2089 1724 2236 1849 2111 1724 2236 1826 2130 1720 2236 1805 2144 1713 2236 1782 2162 1708 2233 1757 2180 1705 2233 1735 2196 1698 2233 1713 2210 1690 2229 1687 2222 1679 % \special{pn 4}% \special{pa 2230 2108}% \special{pa 1864 1742}% \special{fp}% \special{pa 2234 2088}% \special{pa 1886 1742}% \special{fp}% \special{pa 2234 2068}% \special{pa 1910 1742}% \special{fp}% \special{pa 2234 2044}% \special{pa 1928 1740}% \special{fp}% \special{pa 2236 2026}% \special{pa 1950 1740}% \special{fp}% \special{pa 2236 2004}% \special{pa 1972 1740}% \special{fp}% \special{pa 2236 1982}% \special{pa 1990 1736}% \special{fp}% \special{pa 2236 1960}% \special{pa 2012 1736}% \special{fp}% \special{pa 2236 1938}% \special{pa 2030 1732}% \special{fp}% \special{pa 2236 1916}% \special{pa 2052 1732}% \special{fp}% \special{pa 2236 1894}% \special{pa 2072 1728}% \special{fp}% \special{pa 2236 1872}% \special{pa 2090 1724}% \special{fp}% \special{pa 2236 1850}% \special{pa 2112 1724}% \special{fp}% \special{pa 2236 1826}% \special{pa 2130 1720}% \special{fp}% \special{pa 2236 1806}% \special{pa 2144 1714}% \special{fp}% \special{pa 2236 1782}% \special{pa 2162 1708}% \special{fp}% \special{pa 2234 1758}% \special{pa 2180 1706}% \special{fp}% \special{pa 2234 1736}% \special{pa 2196 1698}% \special{fp}% \special{pa 2234 1714}% \special{pa 2210 1690}% \special{fp}% \special{pa 2230 1688}% \special{pa 2222 1680}% \special{fp}% % ELLIPSE 1 0 3 0 % 4 1919 2169 2229 2225 2229 2225 2229 2225 % \special{pn 13}% \special{ar 1920 2170 310 56 0.0000000 6.2831853}% % LINE 2 2 3 0 % 2 2229 1672 2598 1672 % \special{pn 8}% \special{pa 2230 1672}% \special{pa 2598 1672}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 2229 2169 2598 2169 % \special{pn 8}% \special{pa 2230 2170}% \special{pa 2598 2170}% \special{dt 0.045}% % STR 2 0 3 0 % 3 2380 1924 2380 1960 2 0 % {\scriptsize$h$} \put(23.8000,-19.6000){\makebox(0,0)[lb]{{\scriptsize$h$}}}% % VECTOR 2 0 3 0 % 2 2410 1850 2410 1673 % \special{pn 8}% \special{pa 2410 1850}% \special{pa 2410 1674}% \special{fp}% \special{sh 1}% \special{pa 2410 1674}% \special{pa 2390 1740}% \special{pa 2410 1726}% \special{pa 2430 1740}% \special{pa 2410 1674}% \special{fp}% % VECTOR 2 0 3 0 % 2 2410 1990 2410 2168 % \special{pn 8}% \special{pa 2410 1990}% \special{pa 2410 2168}% \special{fp}% \special{sh 1}% \special{pa 2410 2168}% \special{pa 2430 2102}% \special{pa 2410 2116}% \special{pa 2390 2102}% \special{pa 2410 2168}% \special{fp}% \end{picture}% \end{center} \end{document}