\\(a , h\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3057, \\(xyz\\) \u7a7a\u9593\u306e \\(5\\) \u70b9 A \\(( a , a , 0 )\\) , B \\(( -a , a , 0 )\\) ,\r\nC \\(( -a , -a , 0 )\\) , D \\(( a , -a , 0 )\\) , E \\(( 0 , 0 , h )\\) \u3092\u9802\u70b9\u3068\u3059\u308b\u56db\u89d2\u9310\u3092 \\(P\\) \u3068\u3059\u308b.\r\n\\(P\\) \u306e \\(yz\\) \u5e73\u9762\u306b\u3088\u308b\u65ad\u9762\u306e\u5468\u306e\u9577\u3055\u304c \\(1\\) \u3067\u3042\u308b\u3068\u304d, \u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(h\\) \u3092 \\(a\\) \u306e\u5f0f\u3067\u8868\u305b. \u307e\u305f, \\(a\\) \u304c\u53d6\u308a\u5f97\u308b\u5024\u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u7403 \\(S\\) \u306f \\(P\\) \u306e\u3059\u3079\u3066\u306e\u9762\u306b\u63a5\u3057\u3066\u3044\u308b\u3068\u3059\u308b.\r\n\\(a\\) \u304c (1)<\/strong> \u3067\u6c42\u3081\u305f\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(S\\) \u306e\u4f53\u7a4d\u304c\u6700\u5927\u3068\u306a\u308b \\(a\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\u76f4\u65b9\u4f53 \\(Q\\) \u306f \\(1\\) \u3064\u306e\u9762\u304c \\(xy\\) \u5e73\u9762\u4e0a\u306b\u3042\u308a,\r\n\u3059\u3079\u3066\u306e\u9802\u70b9\u304c \\(P\\) \u306e\u8fba\u4e0a\u307e\u305f\u306f\u9762\u4e0a\u306b\u3042\u308b\u3068\u3059\u308b.\r\n\\(a\\) \u3092\u56fa\u5b9a\u3057\u305f\u3068\u304d, \\(Q\\) \u306e\u4f53\u7a4d\u304c\u53d6\u308a\u5f97\u308b\u5024\u306e\u6700\u5927\u5024\u3092 \\(V(a)\\) \u3068\u304a\u304f.\r\n\\(a\\) \u304c (1)<\/strong> \u3067\u6c42\u3081\u305f\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \\(V(a)\\) \u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n2 \\sqrt{a^2 +h^2} +2a & = 1 \\\\\r\na^2 +h^2 & = \\left( \\dfrac{1}{2} -a \\right)^2 \\\\\r\n\\text{\u2234} \\quad h^2 & = \\dfrac{1}{4} -a\r\n\\end{align}\\]\r\n\\(a \\gt 0\\) , \\(h \\gt 0\\) \u306a\u306e\u3067, \\(a\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306f\r\n\\[\r\n\\underline{0 \\lt a \\lt \\dfrac{1}{4}}\r\n\\]\r\n\u307e\u305f\r\n\\[\r\nh = \\underline{\\dfrac{\\sqrt{1 -4a}}{2}}\r\n\\]\r\n (2)<\/strong><\/p>\r\n F \\(( 0 , a , 0 )\\) \u3068\u3057, \\(S\\) \u306e\u4e2d\u5fc3\u3092 P, \\(S\\) \u3068 EF \u306e\u63a5\u70b9\u3092 Q \u3068\u3059\u308b. (3)<\/strong><\/p>\r\n \\(z = t h \\ ( 0 \\lt t \\lt 1 )\\) \u5e73\u9762\u4e0a\u306b \\(Q\\) \u306e\u9762\u304c\u3042\u308b\u3068\u304d, \u3053\u306e\u9762\u306e\u9762\u7a4d\u304c\u6700\u5927\u3068\u306a\u308c\u3070\u3088\u304f, \u3053\u308c\u306f \\(P\\) \u306e\u65ad\u9762\u3068\u4e00\u81f4\u3059\u308b\u3068\u304d\u3067\u3042\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\[\r\n\\text{EF} = \\sqrt{a^2 -a +\\dfrac{1}{4}} = \\dfrac{1}{2} -a\r\n\\]\r\n\\(S\\) \u306e\u534a\u5f84\u3092 \\(r\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n\\text{EP} = h-r \\ , \\ \\text{PQ} = r\r\n\\]\r\n\\(\\triangle \\text{EOF} \\sim \\triangle \\text{EQP}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\left( \\dfrac{\\sqrt{1 -4a}}{2} -r \\right) : r & = \\left( \\dfrac{1}{2} -a \\right) : a \\\\\r\nr \\left( \\dfrac{1}{2} -a \\right) & = a \\left( \\dfrac{\\sqrt{1 -4a}}{2} -r \\right) \\\\\r\n\\text{\u2234} \\quad r & = a \\sqrt{1 -4a}\r\n\\end{align}\\]\r\n\\(S\\) \u306e\u4f53\u7a4d\u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f, \\(r\\) \u304c\u6700\u5927\u3068\u306a\u308b\u3068\u304d\u3067, \\(f(a) = r^2 = a^2 ( 1 -4a )\\) \u3068\u304a\u3044\u3066\u6700\u5927\u3068\u306a\u308b\u3068\u304d\u3092\u8003\u3048\u308c\u3070\u3088\u3044.\r\n\\[\r\nf'(a) = 2a -12a^2 = 2a ( 1 -6a )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(f(a)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} a & (0) & \\cdots & \\dfrac{1}{6} & \\cdots & \\left( \\dfrac{1}{4} \\right) \\\\ \\hline f'(a) & & + & 0 & - & \\\\ \\hline f(a) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(a\\) \u306e\u5024\u306f\r\n\\[\r\na = \\underline{\\dfrac{1}{6}}\r\n\\]\r\n
\r\n\u305d\u306e\u9762\u7a4d\u306f\r\n\\[\r\n( 2a )^2 \\cdot \\dfrac{( h -th )^2}{h^2} = 4a^2 ( 1-t )^2\r\n\\]\r\n\u3053\u306e\u3068\u304d, \\(Q\\) \u306e\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = 4a^2 ( 1-t )^2 \\cdot t h \\\\\r\n& = 2 a^2 \\sqrt{1 -4a} \\cdot \\underline{t (1-t)^2} _ {[1]}\r\n\\end{align}\\]\r\n\\(g(t) = [1]\\) \u3068\u304a\u304f\u3068\r\n\\[\r\ng'(t) = (1-t)^2 -2t (1-t) = (1-t) ( 1 -3t )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(g(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} t & (0) & \\cdots & \\dfrac{1}{3} & \\cdots & (1) \\\\ \\hline g'(t) & & + & 0 & - & \\\\ \\hline g(t) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(g \\left( \\dfrac{1}{3} \\right) = \\dfrac{1}{3} \\left( \\dfrac{2}{3} \\right)^2 = \\dfrac{4}{27}\\) \u306a\u306e\u3067\r\n\\[\r\nV(a) = \\dfrac{8}{27} \\underline{a^2 \\sqrt{1 -4a}} _ {[2]}\r\n\\]\r\n\\(h(a) = [2]^2\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nh'(a) = 4 a^3 -20 a^4 = 4a^3 ( 1 -5a )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(h(a)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u901a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} a & (0) & \\cdots & \\dfrac{1}{5} & \\cdots & \\left( \\dfrac{1}{4} \\right) \\\\ \\hline h'(a) & & + & 0 & - & \\\\ \\hline h(a) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \\(h \\left( \\dfrac{1}{5} \\right) = \\dfrac{1}{5^5}\\) \u306a\u306e\u3067, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f\r\n\\[\r\nV \\left( \\dfrac{1}{5} \\right) = \\dfrac{8}{27} \\sqrt{\\dfrac{1}{5^5}} = \\underline{\\dfrac{8 \\sqrt{5}}{3375}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a , h\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3057, \\(xyz\\) \u7a7a\u9593\u306e \\(5\\) \u70b9 A \\(( a , a , 0 )\\) , B \\(( -a , a , 0 )\\) , C \\(( -a , -a , 0 )\\) , D … \u7d9a\u304d\u3092\u8aad\u3080