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[Title]
403_̐}`

% 蕶
% Ȃ΁C[Level1]ɏ̂̂܂ܖ蕶ƂȂB
[Problem]
̖₢ɓB
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% Level1̖BȉLevel7܂œlB
% 1sڂɂ͏ڍאݒ̃^CgB
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% Ɖ𓚁C𓚂Ɩ͂PsďB
% vZߒꍇ́CƉ𓚂̊ԂɂPsԊuC
% ŏprocessƂsC̎̍svZߒĂB
[Level1]

k500001̒n}ł4.5cm̒́Cۂɂ͉kmB

process
${\rm 4.5cm \times 50000=225000 cm =2250m=2.25km}$

${\rm 2.25km}$

k10001̒n}Ŗʐς$4 {\rm cm^2}$̓yn̎ۂ̒l͉${\rm m^2}$B

process
ʐς$4 {\rm cm^2}$C`Ɖ肷ƕӂ̒$=2cm$B
1000{$2\times 1000=2000{\rm cm=20m}$BāCۂ̖ʐς
$20\times 20=400{\rm m^2}$\\
iʁj
${\rm 4 cm^2 \times 1000^2=4 \times 10^6 cm^2=4 \times 10^2 m^2=400m^2}$

$400 {\rm m^2}$

䂪$2 : 3$2̐`ɂāC̐`̖ʐς$20 {\rm cm^2}$̂ƂC
傫̐`̖ʐς߂B

process
ʐϔ$2^2:3^2=4:9$C$20 \times \displaystyle \frac{9}{4}=45$

$45 {\rm cm^2}$

Op`̒ӂ̒ςɍR{ɂB̂Ƃʐς́iAj{ɂȂB

iAj3

Op`̌`ςɒӂƍꂼR{ɂB̂Ƃʐς́iCj{ɂȂC͂̒́iEj{ɂȂB

iCj9CiEj3

鐳`̏cƉ𓯂䗦ł̂΂Cʐς͂Q{ɂȂBӂ̒́iAj{ɂȂB

process
iAj$x^2=2$

$iAj\sqrt{2}$

钼̂̏cƉƍꂼQ{ɂB̂Ƃ\ʐς́iCj{ɂȂC̐ς́iEj{ɂȂB

process
iCj$2^2=4$CiEj$2^3=8$

iCj4CiEj8

~̗eACBCeς͂Ƃ32$\,l$łB܁CACBƂɐ[$\displaystyle \frac{1}{2}$̂Ƃ܂ŐĂB
Aׂ̐BɓƁCBɂ͂Ɖ$l$邩B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic01.tex}
\end{center}

process
̐[$\displaystyle \frac{1}{2}$ł邩C\\
$=\displaystyle \frac{1}{2}$C
̐ϔ$=\left( \displaystyle \frac{1}{2} \right)^3=\displaystyle \frac{1}{8}$B\\
eς$32\,l$ł邩CĂ鐅̗ʂACBƂ$32 \times \displaystyle \frac{1}{8}=4\,l$B
ׂBɓƁCB̗̐ʂ$8\,l$BāCBɂ͂$32-8=24\,l$B

$24\,l$

}̐lpPQ͑łBP̕\ʐς$108{\rm cm^2}$̂ƂCQ̕\ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic02.tex}
\end{center}

process
$108 \times \left( \displaystyle \frac{5}{3} \right)^2$

$300 \,{\rm cm^2}$

}̐lpPQ͑łBQ̑̐ς$500{\rm \,cm^3}$̂ƂCP̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic02.tex}
\end{center}

process
$500 \times \left( \displaystyle \frac{3}{5} \right)^3$

${\rm 108 \,cm^3}$

}̂悤ɁCȓ̉~${\rm P,\,Q}$B
${\rm P}$${\rm Q}$̑䂪$1:2$łƂC${\rm P}$${\rm Q}$̑̐ϔ$1:\fbox{ L }$ łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic206.tex}
\end{center}

process
u̐ϔvuӂ3̔vC\\$1^3:2^3=1:8$

$8$

}̂悤ɁCpOp`ABC̎ΕAC̓_DCABCBCɐCꂼ̌_ECFƂB
AC $=9\,{\rm cm}$ŎOp`AEDƎOp`DFC̖ʐϔ䂪$1:4$łƂCAD͉̒${\rm cm}$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic208.tex}
\end{center}

process
iʐϔj$=1:4$Ciӂ̔j$=1:2$\\
{\bfiʁj}\,AD $=x$ƂƁCDC $=9-x$B\\
uʐϔvuӂ2̔vC\\
$1:4=x^2:(9-x)^2$\, $4x^2=(9-x)^2$\\
$4x^2=81-18x+x^2$ ,\,$3x^2+18x-81=0$\\
$x^2+6x-27=0$,\, $(x-3)(x+9)=0$ $x=3,\,-9$B
$x>0$C$x=3$

$3 \,{\rm cm}$

}̂悤ɁC̑ȃJbv${\rm P}$C${\rm Q}$C${\rm P}$̍${\rm 6 cm}$C${\rm Q}$̍${\rm 8 cm}$łB
${\rm P}$ɓ邱Ƃ̂ł鐅̑̐ς${\rm 27 \pi cm^3}$łƂC
${\rm Q}$ɓ邱Ƃ̂ł鐅̑̐ς͉${\rm cm^3}$B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic217.tex}
\end{center}

process
$27 \pi \times \left( \displaystyle \frac{8}{6} \right)^3
=27 \pi \times \left( \displaystyle \frac{4}{3} \right)^3
=27 \pi \times \displaystyle \frac{64}{27} $

${\rm 64 \pi cm^3}$

a$1$cm̋${\rm P}$Ɣa$2$cm̋${\rm Q}$B
̂ƂC${\rm P}$Ƌ${\rm Q}$̑̐ϔ$1: \fbox{\,J\,}$

process
u̐ϔvuӂ3̔vC\\$1^3:2^3=1:8$

$8$

}̂悤ɁC̐܊p`C̑$4:5$łB
܊p`${\rm P}$̖ʐς$32 $cm$^2$łƂC܊p`${\rm Q}$̖ʐς͉cm$^2$B\\
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic228.tex}
\end{center}

process
uʐϔvuӂ2̔vC\\
$32 \times \displaystyle \frac{5^2}{4^2}=32 \times \displaystyle \frac{25}{16}=50$

$50\,$cm$^2$

}̂悤ɁCȓ̗̒̂e${\rm P}$C${\rm Q}$B
e${\rm P}$̐[$3$\,cmCe${\rm Q}$̐[$6$\,cmłB
e${\rm P}$ɂςɐāCe${\rm Q}$ɐڂւ邱ƂJԂƁC
ŗe${\rm Q}$ςɂȂ邩B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic234.tex}
\end{center}

process
u̐ϔvuӂ3̔vC\\$1^3:2^3=1:8$

$8$

̐}̎OpABCDƎOpEFGH͑ŁCΉ2ABCEFɂāC\\
AB:EF$=1:3$łB\\
OpABCD̕\ʐς$7 {\rm cm^2}$łƂCOpEFGH̕\ʐς͉${\rm cm^2}$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic238.tex}
\end{center}

process
߂\ʐς$S$ƂƁC\\
$7:S=1^2:3^2=1:9$C$S=7 \times 9=63$

$63 {\rm cm^2}$

̐}̂悤ȉ~̃OXɏȃOXŐꂽƂC
$\displaystyle \frac{1}{4}$̍܂ŐB
̃OXςɂɂ́Cɂɉt̐Kv邩B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic101.tex}
\end{center}

process
傫ȃOXƓ̑́C\\
$1:\displaystyle \frac{1}{4}=4:1$\\
̐ϔ́C\\
$4^3:1^3=64:1$\\
āC$64-1=63t$̐ƂςɂȂB

63t

}̂悤ȑȓ̉~PCQC
P̍2\,cmCQ̍4\,cmłB
~P̑̐ς$6\,\rm{cm}^3$łƂC
傫~Q̑̐ς͉$\rm{cm}^3$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic239.tex}
\end{center}

process
${\rm 6cm^3\times\displaystyle\frac{4^3}{2^3}
=6cm^3\times 2^3=48}$

${\rm 48 cm^3}$

}ɂď~Ɖ̉~̑̐ϔ
1:\fbox{@@}łB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic102.tex}
\end{center}

process
i~̑̐ρjFiƂ̉~̑̐ρj\\
$=1^3:2^3=1:8$\\
i~̑̐ρjFi~̑̐ρj\\
$=1:(8-1)=1:7$

7

}̂悤ȉ~`̗e${\rm 108 cm^3}$̐
$\displaystyle\frac{3}{5}$̍ɂȂB
Ɖ${\rm cm^3}$̐ꂽe͂ςɂȂ邩B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic104.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic104p.tex}
\end{center}
i傫~̑̐ρjFi~̑̐ρj
$=5^3:3^3$ł邩C\\
i傫~̑̐ρj$=108\times \displaystyle\frac{5^3}{3^3}=500$\\
߂̐ς́C\\
$500-108=392$

${\rm 392 \,cm^3}$

$\triangle $ABC$\triangle $PQR͑Ȑ}`ŁC$1:2$łB
$\triangle $ABC̖ʐς$12\, {\rm cm^2}$łƂC$\triangle $PQR̖ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic302.tex}
\end{center}

process
$12\times \displaystyle\frac{2^2}{1^2}=12\times 4=48$

$48 \,{\rm cm^2}$

AƋB͑ȗ̂ŁC$1\,:\,3$łB
A̕\ʐς$4\pi \,{\rm cm^2}$łƂCB̕\ʐς͉${\rm cm^2}$B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic303.tex}
\end{center}

process
$4\pi\times \displaystyle\frac{3^2}{1^2}=4\pi \times 9=36\pi$

$36 \pi {\rm cm^2}$

OpAƎOpB͑ȗ̂ŁC$1\,:\,2$łB
OpȂ̐ς$8 {\rm cm^3}$łƂCOpB̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic304.tex}
\end{center}

process
$8 \times \displaystyle\frac{2^3}{1^3}=64$

$64\,{\rm cm^3}$

}̒AƒB̑$2\,:\,3$łB
A̕\ʐς$200{\rm cm^2}$̂ƂCB̕\ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic307.tex}
\end{center}

process
$200 \times \displaystyle\frac{3^2}{2^2}=450$

$450 {\rm cm^2}$

2̑ȒPCQC$2:3$łB
Q̕\ʐς${\rm 36 cm^2}$
̂ƂCP̕\ʐς߂ȂB

process
$S=36 \times \displaystyle\frac{2^2}{3^2}=16$

${\rm 16 cm^2}$

2̑Ȓ`ECFC$2:5$łB
E̖ʐς${\rm 8 cm^2}$
̂ƂCF̖ʐς߂ȂB

process
$S=8 \times \displaystyle\frac{5^2}{2^2}=50$

${\rm 50 cm^2}$

2̑Ȓ`ECFC$3:4$łB
F̖ʐς${\rm 80 cm^2}$
̂ƂCF̖ʐς߂ȂB

process
$S=80 \times \displaystyle\frac{3^2}{4^2}=45$

${\rm 45 cm^2}$

2̑ȎOpPCQC$2:3$łB
Q̑̐ς${\rm 54 cm^3}$
̂ƂCP̑̐ς߂ȂB

process
$V=54 \times \displaystyle\frac{2^3}{3^3}=16$

${\rm 16 cm^3}$

2̑ȉ~PCQC$1:2$łB
P̑̐ς$3\pi {\rm cm^3}$
̂ƂCQ̑̐ς߂ȂB

process
$V=3\pi \times \displaystyle\frac{2^3}{1^3}=24\pi$

$24\pi {\rm cm^3}$






[Level2]

ʂ̔a3ŁC4̉~̑̐ς߂Bi~$\pi$Ƃj

process
$\displaystyle \frac{1}{3} \times \pi \times 3^2 \times 4=12 \pi$

$12 \pi$

ʂ̔a3ŁC̒5̉~̑̐ς߂Bi~$\pi$Ƃj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic309.tex}
\end{center}

process
O̒藝C~̍4B\\
$\displaystyle \frac{1}{3} \times \pi \times 3^2 \times 4=12 \pi$

$12 \pi$

ʂ̔a4ŁC3̉~̑̐ς߂Bi~$\pi$Ƃj

process
$\displaystyle \frac{1}{3} \times \pi \times 4^2 \times 3=16 \pi$

$16 \pi$

~̓WJ}a$6{\rm cm}$̔~̂ƂC̉~̑̐ς߂iPʁF${\rm cm^3}$jB

process
~̐`̒$(2 \pi \times 6) \times \displaystyle \frac{1}{2}= 6 \pi$B
ʂ̔a$r$ƂƁC$2 \pi r=6 \pi$B$r=3 \,{\rm cm}$B$6 \,{\rm cm}$C$3 \sqrt{3} \,{\rm cm}$BāC̐ς́C\\
$\displaystyle \frac{1}{3} \times \pi E3^2 \times 3 \sqrt{3}$

${\rm 9 \sqrt{3} \pi \,cm^3}$

}̂悤Ȑlp̑̐ς߂iP${\rm cm^3}$jB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic13.tex}
\end{center}

process
$h$ƂƁC$\sqrt{2}^2+h^2=3^2$B$h=\sqrt{7}$B߂̐ς
$\displaystyle \frac{1}{3} \times 2^2 \times \sqrt{7}=\displaystyle \frac{4}{3} \sqrt{7}$

$\displaystyle \frac{4}{3} \sqrt{7}$

}̎ΐ̑`$y$̎ɂP]Ăł闧̂̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic14.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic14p.tex}
\end{center}
}C$\triangle$ CPA]Ăł~͒ʂ̔a$4$C$6$̒~B
̒~̑̐ς\\
$\displaystyle \frac{1}{3} \times \pi \times 4^2 \times 6=32 \pi$B
$\triangle$ CQB]Ăł~͒ʂ̔a$2$C$3$̒~ŁC̐ς\\
$\displaystyle \frac{1}{3} \times \pi E2^2 \times 3 = 4 \pi$B\\
߂̐ς$32 \pi -4 \pi=28 \pi$

$28 \pi$

ʂ̔a$5 {\rm cm}$C̒$13 {\rm cm}$̉~B
̉~̑̐ς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic15.tex}
\end{center}

process
~̍$h$ƂƁC\\
$h=\sqrt{13^2-5^2}=\sqrt{144}=12$B
߂̐ς́C\\
$\displaystyle \frac{1}{3} \times\pi \times 5^2 \times 12=100 \pi$

${\rm 100 \pi cm^3}$

}͒ʐ$50 {\rm cm^2}$C$12 \,{\rm cm}$̐lpłB̐lp̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic16.tex}
\end{center}

process
${\rm \displaystyle \frac{1}{3} \times 50 \times 12=200 cm^3}$

${\rm 200 \, cm^3}$

ʂ̔a$3 {\rm cm}$C̒$6 {\rm cm}$̉~ɂāCƑ̐ς߂ȂB
C~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic17.tex}
\end{center}

process
$1:2:\sqrt{3}$̔䂾C$h=3\sqrt{3}{\rm cm}$B\\
̐=$\displaystyle \frac{1}{3} \times \pi \times 3^2 \times (3\sqrt{3})=9\sqrt{3}$

$h=3\sqrt{3}{\rm cm}C̐=9\sqrt{3}{\rm cm^3}$

10cmCʂ̔a6cm̉~̑̐ς߂ȂBC~$\pi$ƂB

process
$\displaystyle \frac{1}{3} \times \pi \times 6^2 \times 10$

$120 \pi$

12cmCʂ̔a6cm̉~̑̐ς߂ȂBC~$\pi$ƂB

process
$\displaystyle \frac{1}{3} \times \pi \times 6^2 \times 12$

$144 \pi$

}́Cʂ1$3\,{\rm cm}$̐`ŁC
$6 \,{\rm cm}$̎lpłB
̎lp̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic305.tex}
\end{center}

process
$\displaystyle\frac{1}{3} \times 4^2 \times 6=32$

$32\,{\rm cm^3}$

}̂悤ȁC~Ɣgݍ킹̂̑̐ς߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic240.tex}
\end{center}

process
~̑̐ς́C$\displaystyle\frac{1}{3} \,\pi\times3^2\times 4=12\pi\,\rm{cm^3}$\\
̑̐ς́C$\displaystyle\frac{1}{2}\times\displaystyle\frac{4}{3} \,\pi\times 3^3=18\pi\,\rm{cm^3}$

$30\pi\,\rm{cm^3}$

}̂悤ȁC~Ɣgݍ킹̂̕\ʐς߂ȂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic240.tex}
\end{center}

process
~̕\ʐς́C$\pi 5^2 \times \displaystyle\frac{2\pi\times 3}{2\pi \times 5}
=\pi\times 5 \times 3=15\pi\,\rm{cm^2}$\\
̕\ʐς́C$\displaystyle\frac{1}{2}\times 4\pi\times 3^2=18\pi\,\rm{cm^2}$

$33\pi\,\rm{cm^2}$







[Level3]
̕\ʐρC̐
a$5\,{\rm cm}$̋̕\ʐς߂iPʁF${\rm cm^2}$jB

process
̕\ʐ=$4 \pi \times 5^2=100 \pi$

$100\pi$

a3cm̋̑̐ςƕ\ʐς߂ȂBC~$\pi$ƂB

process
$̐V=\displaystyle \frac{4}{3} \pi r^3=\displaystyle \frac{4}{3} \times\pi \times 3^3=36 \pi$\\
$\ʐS=4 \pi r^2=4 \pi \times 3^2=36 \pi$

̐${\rm =36 \pi\,cm^3}$C\ʐ${\rm=36\pi\,cm^2}$

a$2\,{\rm cm}$̔̕\ʐς߂iPʁF${\rm cm^2}$jB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic21.tex}
\end{center}

process
~̖ʐ+$\displaystyle \frac{1}{2} \times$̕\ʐ=$\piE2^2+\displaystyle \frac{1}{2} \times 4\pi \times 2^2$

$12\pi$

a$2\,{\rm cm}$̔̑̐ς߂iPʁF${\rm cm^3}$jB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic21.tex}
\end{center}

process
$\displaystyle \frac{1}{2} \times$̑̐=$\displaystyle \frac{1}{2} \times \displaystyle \frac{4}{3} \times\pi \times 2^3$

$\displaystyle \frac{16}{3} \pi$

a$2\, {\rm cm}$̋̕\ʐςƑ̐ς߂B~$\pi$ƂB

process
$S=4 \pi r^2=4 \pi \times 2^2=16 \pi$\\
$V=\displaystyle \frac{4}{3} \pi r^3=\displaystyle \frac{4}{3}\times \pi\times 2^3=\displaystyle \frac{32}{3} \pi$

$\ʐ=16 \pi\,{\rm cm^2}$C$̐=\displaystyle \frac{32}{3} \pi\,{\rm cm^3}$

ʂ̉~̔a$5 \,{\rm cm}$̉~́C${\rm cm}$̂ƂCa$5{\rm cm}$̋Ɠ̐ςɂȂ邩B

process
~̍$x \,{\rm cm}$ƂƁC$\pi E5^2 \times x =\displaystyle \frac{4}{3} \pi E5^3$

$\displaystyle \frac{20}{3}$

}̔̕\ʐς߂BC~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic25.tex}
\end{center}

process
$(4 \times \pi \times 8^2) \times \displaystyle \frac{1}{2}+\pi \times8^2=128 \pi+64 \pi=192 \pi$

$192\pi \,{\rm cm^2}$

}̔a$3\,{\rm cm}$̔̕\ʐς߂ȂBiC~$\pi$ƂBj
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic26.tex}
\end{center}

process
$(4 \times \pi \times 3^2) \times \displaystyle \frac{1}{2}+\pi \times 3^2=18 \pi+9 \pi=27 \pi$

$27\pi \,{\rm cm^2}$

a$6 \,{\rm cm}$̋̑̐ς߂B

process
$\displaystyle \frac{4}{3} \times \pi\times 6^3=288 \pi$

$288 \pi({\rm cm^3})$

a3cm̋̕\ʐςƓ\ʐς̂́C
1̖ʂ̖ʐς߂ȂBC~$\pi$ƂB

process
̕\ʐς$4\pi \times 3^2$ł邩C
$\displaystyle\frac{4\pi \times 3^2}{6}=6\pi$

$6\pi$








[Level4]
p
ʂ̔a4 cmC8 cm̉~ɁCa4 cm̋傤ǂ܂ĂBƉ~̑̐ς̔߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic22.tex}
\end{center}

process
̑̐ς${\rm \displaystyle \frac{4}{3} \pi \times 4^3=\displaystyle \frac{256}{3} \pi \,cm^3}$B\\
~̑̐ς${\rm \pi \times 4^2 \times 8=128 \pi \,cm^3}$B
āC\\
ȋ̐ρjFi~̑̐ρj$=\displaystyle \frac{256}{3} \pi : 128 \pi=2:3$

ȋ̐ρjFi~̑̐ρj$=\displaystyle \frac{256}{3} \pi : 128 \pi=2:3$

ʂ̔a4 cmC8 cm̉~ɁCa4 cm̋傤ǂ܂ĂB̕\ʐςƉ~̑ʐς̔߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic22.tex}
\end{center}

process
̕\ʐς$4 \times \pi \times 4^2=64 \pi \,{\rm cm^2}$B\\
~̑ʐς$(2 \times \pi \times 4)\times 8=64 \pi \,{\rm cm^2}$B\\
āC\\
i̕\ʐρjFi~̑ʐρj\\
$=64 \pi:64 \pi=1:1$

i̕\ʐρjFi~̑ʐρj$=1:1$

}̎ΐ̕𒼐$l$ƂĂP]Ƃɂł闧̂ɂđ̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic23.tex}
\end{center}

process
ABP]ƁCa6cm̔łB̑̐ς߂ƁC\\
$\left( \displaystyle \frac{4}{3} \times \pi \times 6^3 \right) \times \displaystyle \frac{1}{2}=144 \pi \,{\rm cm^3}$B\\
DCP]ƁCa3cm̔łB̑̐ς߂ƁC\\
$\left( \displaystyle \frac{4}{3} \times \pi \times 3^3 \right) \times \displaystyle \frac{1}{2}=18 \pi \,{\rm cm^3}$B\\
āC߂̐ς$144 \pi-18 \pi=126 \pi \,cm^3$

$126 \pi \,{\rm cm^3}$

1ӂ̒1ł鐳ʑ̂̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic103.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic103p.tex}
\end{center}
${\rm BG=\displaystyle\frac{1}{2}\times BD=\displaystyle\frac{1}{2}\times BC\times \sqrt{2}=\displaystyle\frac{\sqrt{2}}{2}}$\\
${\rm AG=\sqrt{1^2-\left( \displaystyle\frac{\sqrt{2}}{2} \right)^2}}=\displaystyle\frac{1}{\sqrt{2}}$\\
߂̐ς$2\times \displaystyle\frac{1}{3}\times 1^2 \times \displaystyle\frac{1}{\sqrt{2}}
=\displaystyle\frac{\sqrt{2}}{3}$

$\displaystyle\frac{\sqrt{2}}{3}$

}̂悤ɁC1ӂ̒12C̐ς$144\sqrt{2}$̐lʑ̂ĈS̖ʂׂĂɐڂ鋅ƂC̔aƋ̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic24.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic24p.tex}
\end{center}
lʑ̂C̒S𒸓_ƂS̍ȎOp킳̂Ƃ݂B̔a$r$ƂƁC
$\left \{ \displaystyle \frac{1}{3} \times \left( \displaystyle \frac{1}{2} \times 12 \times 6 \sqrt{3} \right) \times r \right \} \times 4=144 \sqrt{2}$B
$\sqrt{3} r=3 \sqrt{2}$B$r=\sqrt{6}$B
̑̐ς \\
$\displaystyle \frac{4}{3} \pi (\sqrt{6})^3=8 \sqrt{6} \pi$

a$\sqrt{6}$C̐ς$8 \sqrt{6} \pi$

}̂悤ȓWJ}gݗĂƂɂł
1ӂ̒6̐lp̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic105.tex}
\end{center}

process
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic105p.tex}
\end{center}
${\rm BF:BC}=1:\sqrt{2}$C
${\rm BF}=\displaystyle\frac{6}{\sqrt{2}}$\\
${\rm AF}
=\sqrt{6^2-\left( \displaystyle\frac{6}{\sqrt{2}} \right)^2}\\
=6\sqrt{1-\left( \displaystyle\frac{1}{\sqrt{2}} \right) ^2}
=\displaystyle\frac{6}{\sqrt{2}}$\\
߂̐ς́C\\
$\displaystyle\frac{1}{3} \times 6 \times 6 \times \displaystyle\frac{6}{\sqrt{2}}=36\sqrt{2}$

$36\sqrt{2}$





[Level5]
~COp̕\ʐρC̐
}̎Op̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic306.tex}
\end{center}

process
$\left( \displaystyle\frac{1}{2}\times 6\times  3\right)\times 5=45$

$45\,{\rm cm^3}$

̉~̑̐ς߂B
~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic301.tex}
\end{center}

process
$(\pi \times 20^2)\times 60$

$24000\pi \, {\rm cm^3}$

̉~̕\ʐς߂B
~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic301.tex}
\end{center}

process
$(\pi \times 20^2)\times 2+(2\pi \times 20)\times 60=800\pi+2400\pi=3200\pi$

$3200\pi \, {\rm cm^2}$

}̎Op̑̐ς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic306.tex}
\end{center}

process
$\left( \displaystyle\frac{1}{2}\times 6\times  3\right)\times 5=45$

$45\,{\rm cm^3}$

}̉~̕\ʐς߂B
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic308.tex}
\end{center}

process
$(\pi \times 6^2)\times 2+(2\pi \times 6) \times 12\\
=72\pi +144\pi=216\pi$

$216\pi \,{\rm cm^2}$

̉~̑̐ς߂B
~$\pi$ƂB
\begin{center}
\input{D:/texlive/2018/bin/win32/mathtex/403_̐}`/pic310.tex}
\end{center}

process
$(\pi \times 2^2)\times 5$

$20\pi \, {\rm cm^3}$




[Level6]


[Level7]


[EOF]
% t@C̍Ōɂ
