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修改13492，13493题目

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WangWeiye 2023-05-19 17:27:25 +08:00
parent 3319347332
commit 7b4c8ed2bf
2 changed files with 7 additions and 5 deletions
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2
工具/latex界面修改题目内容.py
View File

@ -1,6 +1,6 @@
import os,re,json
"""这里编辑题号(列表)后将在vscode中打开窗口, 编辑后保存关闭"""
problems = "15073"
problems = "13492,13493"
editor = "王伟叶"
def generate_number_set(string,dict):

10
题库0.3/Problems.json
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@ -355516,7 +355516,7 @@
},
"013492": {
"id": "013492",
"content": "已知集合$A=\\{x | x^2-3 x+2 \\leq 0\\}$, $B=\\{x | \\dfrac{x-a}{x+2}>0, a>0\\}$, 若``$x \\in A$''是``$x \\in$$B$''的充分非必要条件, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$0<a<1$}{$a \\geq 2$}{$1<a<2$}{$a \\geq 1$}",
"content": "已知集合$A=\\{x | x^2-3 x+2 \\leq 0\\}$, $B=\\{x | \\dfrac{x-a}{x+2}>0\\}$, 其中常数$a>0$. 若``$x \\in A$''是``$x \\in$$B$''的充分非必要条件, 则$a$的取值范围是\\bracket{20}.\n\\fourch{$0<a<1$}{$a \\geq 2$}{$1<a<2$}{$a \\geq 1$}",
"objs": [],
"tags": [
"第一单元"
@ -355528,7 +355528,8 @@
"usages": [],
"origin": "2022版双基百分百",
"edit": [
"20230123\t王伟叶"
"20230123\t王伟叶",
"20230519\t王伟叶"
],
"same": [],
"related": [],
@ -355538,7 +355539,7 @@
},
"013493": {
"id": "013493",
"content": "刘徽在他的《九章算术注》中提出一个独特的方法来计算球体的体积: 他不直接给出球体的体积, 而是先计算另一个叫``牟合方盖''的立体的体积. 刘徽通过计算, ``牟合方盖''的体积与球的体积之比应为$\\dfrac{4}{\\pi}$. 后人导出了``牟\n合方盖''的$\\dfrac{1}{8}$体积计算公式, 即$\\dfrac{1}{8} V_{\\text {牟 }}=r^3-V_{\\text{方盖差}}$, $r$为球的半径, 也即正方形的棱长均为$2 r$, 从而计算出$V_{\\text {球}}=\\dfrac{4}{3} \\pi r^3$. 记所有棱长都为$r$的正四棱锥的体积为$V_{\\text{正}}$, 棱长为$2r$的正方形的方盖差的体积为$V_{\\text{方盖差}}$, 则$\\dfrac{V_{\\text{方盖差}}}{V_{\\text{正}}}=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A1);\n\\draw (B) ++ (0,\\l,0) coordinate (B1);\n\\draw (C) ++ (0,\\l,0) coordinate (C1);\n\\draw (D) ++ (0,\\l,0) coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(B1)$) -- ($(C1)!0.5!(D1)$) ($(A1)!0.5!(D1)$) -- ($(B1)!0.5!(C1)$);\n\\draw [dashed] ($(A)!0.5!(D)$) -- ($(B)!0.5!(C)$);\n\\draw [domain = 0:360] plot ({1+cos(\\x)},{1+sin(\\x)},0);\n\\draw [domain = 0:360] plot (2,{1+cos(\\x)},{-1+sin(\\x)});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{2}}{2}$}{$\\sqrt{2}$}{$\\sqrt{3}$}",
"content": "刘徽在他的《九章算术注》中提出一个独特的方法来计算球体的体积: 他不直接给出球体的体积, 而是先计算另一个叫``牟合方盖''的立体的体积. 刘徽通过计算, ``牟合方盖''的体积与球的体积之比应为$\\dfrac{4}{\\pi}$. 后人导出了``牟\n合方盖''的$\\dfrac{1}{8}$体积计算公式, 即$\\dfrac{1}{8} V_{\\text {牟 }}=r^3-V_{\\text{方盖差}}$, $r$为球的半径, 也即正方体的棱长均为$2 r$, 从而计算出$V_{\\text {球}}=\\dfrac{4}{3} \\pi r^3$. 记所有棱长都为$r$的正四棱锥的体积为$V_{\\text{正}}$, 棱长为$2r$的正方体的方盖差的体积为$V_{\\text{方盖差}}$, 则$\\dfrac{V_{\\text{方盖差}}}{V_{\\text{正}}}=$\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) coordinate (A);\n\\draw (A) ++ (\\l,0,0) coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) coordinate (C);\n\\draw (A) ++ (0,0,-\\l) coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) coordinate (A1);\n\\draw (B) ++ (0,\\l,0) coordinate (B1);\n\\draw (C) ++ (0,\\l,0) coordinate (C1);\n\\draw (D) ++ (0,\\l,0) coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(B1)$) -- ($(C1)!0.5!(D1)$) ($(A1)!0.5!(D1)$) -- ($(B1)!0.5!(C1)$);\n\\draw [dashed] ($(A)!0.5!(D)$) -- ($(B)!0.5!(C)$);\n\\draw [domain = 0:360] plot ({1+cos(\\x)},{1+sin(\\x)},0);\n\\draw [domain = 0:360] plot (2,{1+cos(\\x)},{-1+sin(\\x)});\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac{1}{2}$}{$\\dfrac{\\sqrt{2}}{2}$}{$\\sqrt{2}$}{$\\sqrt{3}$}",
"objs": [],
"tags": [
"第六单元"
@ -355550,7 +355551,8 @@
"usages": [],
"origin": "2022版双基百分百",
"edit": [
"20230123\t王伟叶"
"20230123\t王伟叶",
"20230519\t王伟叶"
],
"same": [],
"related": [],